p^4阶有限环序列以及R81的编号

特征为27、加法群为C_27×C_3的81阶环分配编号空间6~35、515（已找到的上限值）
特征为9、加法群为C_9×C_9的81阶环分配编号空间36~148（已找到的上、下限值）
特征为9、加法群为C_9×C_3×C_3的81阶环分配编号空间149~299、430~514、516~520、522~526（已找到的上、下限值）
特征为3的81阶环分配编号空间300~429、521（已找到的下限值）

Number of rings with additive group (Z/nZ)^2.
1, 8, 8, 66, 8, 64, 8, 301, 175, 64, 8, 528, 8, 64, 64
Number of commutative rings with additive group (Z/nZ)^2.
1, 6, 6, 28, 6, 36, 6, 79, 35, 36, 6, 168, 6, 36, 36
加群(4,4)型的16阶结合环：非交换环38个，交换环28个，合计66个【已找到35+27=62个：R16_52~R16_113】
加群(8,8)型的64阶结合环：非交换环222个，交换环79个，合计301个
加群(9,9)型的81阶结合环：非交换环140个，交换环35个，合计175个【已找到83+30=113个】

赵嗣元：域F_p上四维结合代数的同构分类(2001年发表于上海师范大学学报(自然科学版))
加群为初等p群即(p,p,p,p)型的p^4阶结合环的同构分类【p=2的情形有重复的】：
非幂零环2p+96-(-1)^p个，幂零环19+(11/2)p-(1+11(-1)^p)/4+[1-{(p-1)/3}]+[1-{(p-1)/4}]个
p=2时，非幂零环99个，幂零环27个，合计126个【已找到93+27=120个：38~39、273~390，16阶非幂零环已找到93个：38~39、273、278~281、284、285、289、293、295~300、302、305~332、335、336、338~342、344、346~356、358~360、364~366、368~378、380~390】
p=3时，非幂零环103个，幂零环19+33/2+5/2=38个，合计141个【已找到131个】

定理1：域F_p上四维非幂零结合代数亦即加群为初等p-群的p^4阶非幂零结合环恰有2p+96-(-1)^p个同构类。下面的表1选出了一个全体代表团。
证法已体现在分类清单表1之中：先按根基的维数d1来分。这里d1∈{0,1,2,3},根基都是较低阶的幂零环，其分类与模范代表都是已知的；
次按根基的幂零指数r∈{2,3,···,d1+1}来分；
再因代数对根基的半单商代数只是有限域或有限域的直和，在基域扩张下，仍是域或域的直和，故是可离的。
按Wedderburn主定理：代数是一半单子代数S（与商代数同构）与根基N之和，且作为子空间之和是直和（作为子代数未必是直和），作为理想的根基N既是左S-模，又是右-S模，即N的承载空间既是S的左乘表示d_L的表示空间，又是S的右乘表示d_R的表示空间。
d_L的核I_L是S之一理想。它从左面零化N，即I_LN=(0)。同样，d_R的核I_R也是S之一理想。它从右面零化N，即NI_R=(0)。
于是就可按(I_L,I_R)的所有可能情形来分类，此时必须舍弃重复出现的同构者。
N代表幂零环，R代表非幂零环，下标均系加群类型，上标中r表根基的幂零指数，l标编号，d_j是N^j的维数(j=1,2)
d1=0,r=1有6种：M_2(F_p),F_p^4,F_p^3+F_p,2F_p^2,F_p^2+2F_p,4F_p
d1=1,r=2有10种：
d1=2,r=2、3有29种：
d1=3,r=2、3、4有20+(23+p+3+(1-(-1)^p))/2+p-1+1+(1-(-1)^p))/2)+4=51+2p-(-1)^p种：

R16_309=R[(3,3,1),(1,1,1,1)]
R16_375=R[(3,3,2),(1,1,1,1)]

赵嗣元：加群(p^(k-1),p)型的p^k(k>3)阶结合环的同构分类(1990年发表于数学研究与评论杂志)
p>2：非幂零环k+6个，幂零环(p+1)(3k-7)+8个，合计(3k-7)p+(4k+7)个
p=2：非幂零环k+5个，幂零环4(2k-3)个，合计9k-7个
推论：
加群(8,2)型的16阶结合环：非幂零环9个，幂零环20个，合计29个【已全部找到：R16_6~R16_34】
加群(16,2)型的32阶结合环：非幂零环10个，幂零环28个，合计38个【已找到34个：R32_7~R32_40】
加群(27,3)型的81阶结合环：非幂零环10个，幂零环28个，合计38个【已找到10+21=31个：R81_6~R81_35、R81_515】
加群(81,3)型的243阶结合环：非幂零环11个，幂零环40个，合计51个【已找到29个：R243_7~R243_35】

侯影：有主幂等元的p^4环(2002年吉林大学硕士学位论文，导师：杜现昆)
有非平凡主幂等元的p^4阶不可分环共有35(p≠2)或33(p=2)个
p^2阶结合环
文献1[1938年]
文献2[1964年]
文献3[1969年]
文献7[1985年]:特征为p的p^4阶单式环的个数有20个。
文献8[1994年]:p^4阶非交换幺环的个数是：当p=2有13个，当p>2时有2p+12个[与文献9给出的结果不一致]。
文献9[1995年]
文献10[2000年]
文献11[2000年]给出了p^n(n<=5)阶不可分幺环的分类：
当n=1时有1个；
当n=2时有3个；
当n=3时有8(p≠2)或7(p=2)个；
当n=4时有2p+35(p≠2且p≡1(3))或2p+33(p≠2且p!≡1(3))或33(p=2)个；
当n=5时有2p^2+16p+97(p=2)或2p^2+22p+128(p=3)或2p^2+22p+142(p≡1(12))或2p^2+22p+130(p≡5(12))或2p^2+22p+138(p≡7(12))或2p^2+22p+126(p≡11(12))。
文献12[1956年]
p^3阶结合环
文献4:Associative rings of order p^3,Robert Gilmer, Joe Mott发表于1973年，结论有错误
文献4[1973年]给出的p^3阶环的个数是：当p=2有59个，当p>2时有4p+48个。
可分解环Decomposable有20=4+16、20=4+16个,其中加群型(p^2,p)有16个，加群型(p,p,p)有4个
不可分解环Indecomposable有39、4p+28个,其中加群型(p^3)有4个，加群型(p^2,p)有17、3p+13个，加群型(p,p,p)有18、p+11个
加群型(p^2,p)不可分解幺环有2、3个：
Z[x]/(4,2x,x^2)，Z[x]/(4,2x,x^2-2)
Z[x]/(p^2,px,x^2)，Z[x]/(p^2,px,x^2-p)，Z[x]/(p^2,px,x^2-kp)
加群型(p^2,p)不可分解非交换非幺环有5、2p+3个：
{{{a,b},{0,0}}|a∈Z/(p^2),b∈pZ/(p^2)}，{{{0,b},{0,a}}|a∈Z/(p^2),b∈pZ/(p^2)}
加群型(p,p,p)不可分解交换幺环有3、3个：
GF(p^3)，Z[x]/(p,x^3)，Z[x,y]/(p,x^2,xy,y^2)
文献6:V. G. Antipkin and V. P. Elizarov, Rings of order p^3, Sib. Math. J. vol 23 no 4 (1982) pp 457-464, MR0668331 (84d:16025).
a(8) = 52, a(p^3) = 3p + 50 if p is an odd prime.
文献5[1975年]、文献6[1982年]给出的p^3阶环的个数是：当p=2有52个，当p>2时有3p+50个。
文献5、文献6给出的非幂零环分别是29、18个【27阶非幂零环已找到31个：3、7、13~15、17、20~22、26~28、30、32、33、35~38、41~47、49~52、59】。
三者给出的幺环的个数是一致的。

推论：
16阶不可分幺环有33个，16阶非交换幺环有13个
32阶不可分幺环有137个
81阶不可分幺环有39个，81阶非交换幺环有18个
243阶不可分幺环有212个

16阶可分幺环有17个：6,104,225,226,227,248,254,302,323,332,380,382,384,386,387,388,389，
16阶不可分幺环有33个：3,7,32,39,50,52,92,105,106,110,119,123,124,170,179,191,192,193,205,284,293,295,297,299,300,364,366,371,374,377,383,385,390，
16阶非交换幺环有13个：39,119,123,191,192,284,295,297,300,364,371,374,380，
81阶可分幺环有18个(已全部找到)：6,104,166,216,227,244,270,315,321,332,369,380,385,388,394,402,412,510,
81阶不可分幺环有39个(已找到35个)：3,7,31,33,105,106,110,148,149,205,256,300,302,307,356,374,383,390,393,395,397,405,419,420,421,426,427,429,456,492,497,506,507,512,522
81阶非交换幺环有18个(已找到14个)：256,300,356,374,380,395,397,405,419,426,427,429,492,506,

16阶交换环已找到161种:
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,22,24,26,27,29,30,31,32,43,46,50,52,53,64,70,71,75,79,84,91,92,93,95,96,97,98,99,100,103,104,105,106,107,109,110,111,112,113,114,124,131,132,134,136,138,140,147,148,162,167,170,173,179,180,181,183,188,189,193,197,200,205,206,207,208,210,211,212,213,214,216,217,218,219,220,221,224,225,226,227,229,230,234,235,237,238,239,243,244,248,249,250,251,252,253,254,257,260,267,270,286,287,293,299,301,302,303,304,305,306,307,314,315,316,319,320,323,324,325,326,329,331,332,333,334,338,339,345,346,348,361,362,366,367,377,382,383,384,385,386,387,388,389,390,
16阶非交换环已找到225种:
18,19,20,21,23,25,28,33,34,38,39,40,41,42,44,45,47,48,49,54,55,56,57,58,59,60,61,62,63,65,66,67,68,69,72,73,74,76,77,78,80,81,82,83,85,86,87,88,89,90,94,101,102,108,115,116,117,118,119,120,121,122,123,125,126,127,128,129,130,133,135,137,139,141,142,143,144,145,146,149,150,151,152,153,154,155,156,157,158,159,160,161,163,164,165,166,168,169,171,172,174,175,176,177,178,182,184,185,186,187,190,191,192,194,195,196,198,199,201,202,203,204,209,215,222,223,228,231,232,233,236,240,241,242,245,246,247,255,256,258,259,261,262,263,264,265,266,268,269,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,288,289,290,291,292,294,295,296,297,298,300,308,309,310,311,312,313,317,318,321,322,327,328,330,335,336,337,340,341,342,343,344,347,349,350,351,352,353,354,355,356,357,358,359,360,363,364,365,368,369,370,371,372,373,374,375,376,378,379,380,381,
16阶幺环已找到50种【1+3+6+15+25】:
3,6,7,32,39,50,52,92,104,105,106,110,119,123,124,170,179,191,192,193,205,225,226,227,248,254,284,293,295,297,299,300,302,323,332,364,366,371,374,377,380,382,383,384,385,386,387,388,389,390,

81阶交换环已找到171种:
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,22,24,26,27,28,29,30,31,33,42,45,53,66,70,71,75,79,84,91,93,95,96,97,98,99,100,104,105,106,107,109,110,111,112,113,115,121,122,148,149,156,160,161,162,163,166,169,179,180,185,186,188,191,192,193,194,195,196,200,205,206,210,211,212,214,216,217,218,219,220,221,227,228,230,232,236,238,243,244,248,249,250,251,254,255,257,260,270,274,276,292,293,296,301,302,303,305,306,307,311,314,315,321,323,324,325,326,329,331,332,333,334,338,339,340,345,347,361,364,369,377,383,385,387,388,389,390,393,394,399,402,403,412,414,420,421,425,441,456,459,461,465,477,479,481,496,497,507,508,509,510,512,515,522
81阶非交换环已找到355种:
18,19,20,21,23,25,32,34,35,36,37,38,39,40,41,43,44,46,47,48,49,50,51,52,54,55,56,57,58,59,60,61,62,63,64,65,67,68,69,72,73,74,76,77,78,80,81,82,83,85,86,87,88,89,90,92,94,101,102,103,108,114,116,117,118,119,120,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,150,151,152,153,154,155,157,158,159,164,165,167,168,170,171,172,173,174,175,176,177,178,181,182,183,184,187,189,190,197,198,199,201,202,203,204,207,208,209,213,215,222,223,224,225,226,229,231,233,234,235,237,239,240,241,242,245,246,247,252,253,256,258,259,261,262,263,264,265,266,267,268,269,271,272,273,275,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,294,295,297,298,299,300,304,308,309,310,312,313,316,317,318,319,320,322,327,328,330,335,336,337,341,342,343,344,346,348,349,350,351,352,353,354,355,356,357,358,359,360,362,363,365,366,367,368,370,371,372,373,374,375,376,378,379,380,381,382,384,386,391,392,395,396,397,398,400,401,404,405,406,407,408,409,410,411,413,415,416,417,418,419,422,423,424,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,442,443,444,445,446,447,448,449,450,451,452,453,454,455,457,458,460,462,463,464,466,467,468,469,470,471,472,473,474,475,476,478,480,482,483,484,485,486,487,488,489,490,491,492,493,494,495,498,499,500,501,502,503,504,505,506,511,513,514,516,517,518,519,520,521,523,524,525,526
81阶幺环已找到53种【1+4+5+16+27】:
3,6,7,31,33,104,105,106,110,148,149,166,205,216,227,244,256,270,300,302,307,315,321,332,356,369,374,380,383,385,388,390,393,394,395,397,402,405,412,419,420,421,426,427,429,456,492,497,506,507,510,512,522

Rpppp(1,2)=R16_1,Rpppp(1,3)=R81_1
Rpppp(2,2)=R16_2,Rpppp(2,3)=R81_2
Rpppp(3,2)=R16_3,Rpppp(3,3)=R81_3
Rpppp(4,2)=R16_4,Rpppp(4,3)=R81_4
Rpppp(5,2)=R16_5,Rpppp(5,3)=R81_5
Rpppp(7,2)=R16_7,Rpppp(7,3)=R81_7
Rpppp(8,2)=R16_8,Rpppp(8,3)=R81_8
Rpppp(9,2)=R16_9,Rpppp(9,3)=R81_9
Rpppp(10,2)=R16_10,Rpppp(10,3)=R81_10
Rpppp(11,2)=R16_11,Rpppp(11,3)=R81_11
Rpppp(12,2)=R16_12,Rpppp(12,3)=R81_12
Rpppp(13,2)=R16_13,Rpppp(13,3)=R81_16
Rpppp(14,2)=R16_14,Rpppp(14,3)=R81_27
Rpppp(15,2)=R16_15,Rpppp(15,3)=R81_30
Rpppp(16,2)=R16_16,Rpppp(16,3)=R81_26
Rpppp(17,2)=R16_17,Rpppp(17,3)=R81_17
Rpppp(18,2)=R16_18,Rpppp(18,3)=R81_18
Rpppp(19,2)=R16_19,Rpppp(19,3)=R81_19
Rpppp(20,2)=R16_20,Rpppp(20,3)=R81_20
Rpppp(21,2)=R16_21,Rpppp(21,3)=R81_21
Rpppp(22,2)=R16_22,Rpppp(22,3)=R81_22
Rpppp(23,2)=R16_23,Rpppp(23,3)=R81_23
Rpppp(24,2)=R16_24,Rpppp(24,3)=R81_32
Rpppp(25,2)=R16_25,Rpppp(25,3)=R81_25
Rpppp(26,2)=R16_26,Rpppp(26,3)=R81_26、R81_34
Rpppp(27,2)=R16_27,Rpppp(27,3)=R81_17
Rpppp(29,2)=R16_29,Rpppp(29,3)=R81_14
Rpppp(30,2)=R16_30,Rpppp(30,3)=R81_28
Rpppp(31,2)=R16_31,Rpppp(31,3)=R81_15、R81_29
Rpppp(32,2)=R16_32,Rpppp(32,3)=R81_31、R81_33

Rpppp(38,2)=R16_38,Rpppp(38,3)=R81_502
Rpppp(39,2)=R16_39,Rpppp(39,3)=R81_419
Rpppp(40,2)=R16_40,Rpppp(40,3)=R81_155
Rpppp(42,2)=R16_42,Rpppp(42,3)=R81_451、R81_453
Rpppp(43,2)=R16_43,Rpppp(43,3)=R81_163
Rpppp(44,2)=R16_44,Rpppp(44,3)=R81_453
Rpppp(46,2)=R16_46,Rpppp(46,3)=R81_257
Rpppp(47,2)=R16_47,Rpppp(47,3)=R81_433
Rpppp(48,2)=R16_48,Rpppp(48,3)=R81_232
Rpppp(49,2)=R16_49,Rpppp(49,3)=R81_153

Rpppp(53,2)=R16_53,Rpppp(53,3)=R81_70
Rpppp(54,2)=R16_54,Rpppp(54,3)=R81_78、R81_138
Rpppp(55,2)=R16_55,Rpppp(55,3)=R81_37、R81_87
Rpppp(56,2)=R16_56,Rpppp(56,3)=R81_56
Rpppp(57,2)=R16_57,Rpppp(57,3)=R81_80、R81_87
Rpppp(58,2)=R16_58,Rpppp(58,3)=R81_48、R81_81
Rpppp(59,2)=R16_59,Rpppp(59,3)=R81_116
Rpppp(60,2)=R16_60,Rpppp(60,3)=R81_36
Rpppp(61,2)=R16_61,Rpppp(61,3)=R81_65、R81_63
Rpppp(62,2)=R16_62,Rpppp(62,3)=R81_38
Rpppp(63,2)=R16_63,Rpppp(63,3)=R81_46
Rpppp(64,2)=R16_64,Rpppp(64,3)=R81_130、R81_137、R81_66
Rpppp(65,2)=R16_65,Rpppp(65,3)=R81_74、R81_142
Rpppp(66,2)=R16_66,Rpppp(66,3)=R81_77、R81_135
Rpppp(67,2)=R16_67,Rpppp(67,3)=R81_67
Rpppp(68,2)=R16_68,Rpppp(68,3)=R81_68
Rpppp(69,2)=R16_69,Rpppp(69,3)=R81_69、R81_54
Rpppp(70,2)=R16_70,Rpppp(70,3)=R81_103、R81_100
Rpppp(71,2)=R16_71,Rpppp(71,3)=R81_71、R81_145
Rpppp(72,2)=R16_72,Rpppp(72,3)=R81_72、R81_41
Rpppp(73,2)=R16_73,Rpppp(73,3)=R81_73、R81_60、R81_117
Rpppp(74,2)=R16_74,Rpppp(74,3)=R81_74、R81_124
Rpppp(75,2)=R16_75,Rpppp(75,3)=R81_75、R81_42
Rpppp(76,2)=R16_76,Rpppp(76,3)=R81_76、R81_144
Rpppp(77,2)=R16_77,Rpppp(77,3)=R81_46、R81_41
Rpppp(78,2)=R16_78,Rpppp(78,3)=R81_78、R81_134、R81_43
Rpppp(79,2)=R16_79,Rpppp(79,3)=R81_79
Rpppp(80,2)=R16_80,Rpppp(80,3)=R81_74、R81_131、R81_141
Rpppp(81,2)=R16_81,Rpppp(81,3)=R81_64、R81_120
Rpppp(82,2)=R16_82,Rpppp(82,3)=R81_82
Rpppp(83,2)=R16_83,Rpppp(83,3)=R81_83、R81_72
Rpppp(84,2)=R16_84,Rpppp(84,3)=R81_84
Rpppp(85,2)=R16_85,Rpppp(85,3)=R81_85、R81_119
Rpppp(86,2)=R16_86,Rpppp(86,3)=R81_86
Rpppp(87,2)=R16_87,Rpppp(87,3)=R81_88
Rpppp(88,2)=R16_88,Rpppp(88,3)=R81_35
Rpppp(89,2)=R16_89,Rpppp(89,3)=R81_89、R81_114
Rpppp(90,2)=R16_90,Rpppp(90,3)=R81_51
Rpppp(91,2)=R16_91,Rpppp(91,3)=R81_91
Rpppp(92,2)=R16_92,Rpppp(92,3)=R81_104
Rpppp(93,2)=R16_93,Rpppp(93,3)=R81_93
Rpppp(94,2)=R16_94,Rpppp(94,3)=R81_94、R81_140
Rpppp(95,2)=R16_95,Rpppp(95,3)=R81_95
Rpppp(96,2)=R16_96,Rpppp(96,3)=R81_96
Rpppp(97,2)=R16_97,Rpppp(97,3)=R81_97
Rpppp(98,2)=R16_98,Rpppp(98,3)=R81_98
Rpppp(99,2)=R16_99,Rpppp(99,3)=R81_99
Rpppp(100,2)=R16_100,Rpppp(100,3)=R81_84、R81_122

Rpppp(101,2)=R16_101,Rpppp(101,3)=R81_101
Rpppp(102,2)=R16_102,Rpppp(102,3)=R81_102
Rpppp(103,2)=R16_103,Rpppp(103,3)=R81_98
Rpppp(104,2)=R16_104,Rpppp(104,3)=R81_104
Rpppp(105,2)=R16_105,Rpppp(105,3)=R81_105
Rpppp(106,2)=R16_106,Rpppp(106,3)=R81_106
Rpppp(107,2)=R16_107,Rpppp(107,3)=R81_107
Rpppp(108,2)=R16_108,Rpppp(108,3)=R81_108
Rpppp(109,2)=R16_109,Rpppp(109,3)=R81_109
Rpppp(110,2)=R16_110,Rpppp(110,3)=R81_110
Rpppp(111,2)=R16_111,Rpppp(111,3)=R81_111
Rpppp(112,2)=R16_112,Rpppp(112,3)=R81_112
Rpppp(113,2)=R16_113,Rpppp(113,3)=R81_113
Rpppp(114,2)=R16_114,Rpppp(114,3)=R81_195
Rpppp(117,2)=R16_117,Rpppp(117,3)=R81_317、R81_468
Rpppp(119,2)=R16_119,Rpppp(119,3)=R81_497
Rpppp(125,2)=R16_125,Rpppp(125,3)=R81_290、R81_291
Rpppp(126,2)=R16_126,Rpppp(126,3)=R81_495、R81_278
Rpppp(127,2)=R16_127,Rpppp(127,3)=R81_471
Rpppp(128,2)=R16_128,Rpppp(128,3)=R81_451
Rpppp(129,2)=R16_129,Rpppp(129,3)=R81_466
Rpppp(130,2)=R16_130,Rpppp(130,3)=R81_294
Rpppp(131,2)=R16_131,Rpppp(131,3)=R81_185
Rpppp(132,2)=R16_132,Rpppp(132,3)=R81_186
Rpppp(133,2)=R16_133,Rpppp(133,3)=R81_283
Rpppp(134,2)=R16_134,Rpppp(134,3)=R81_296
Rpppp(135,2)=R16_135,Rpppp(135,3)=R81_284
Rpppp(136,2)=R16_136,Rpppp(136,3)=R81_293
Rpppp(137,2)=R16_137,Rpppp(137,3)=R81_289
Rpppp(138,2)=R16_138,Rpppp(138,3)=R81_192
Rpppp(138,2)=R16_138,Rpppp(138,3)=R81_285、R81_291
Rpppp(139,2)=R16_139,Rpppp(139,3)=R81_288、R81_280
Rpppp(140,2)=R16_140,Rpppp(140,3)=R81_191、R81_188、R81_478、R81_147
Rpppp(141,2)=R16_141,Rpppp(141,3)=R81_287、R81_281
Rpppp(142,2)=R16_142,Rpppp(142,3)=R81_287、R81_288
Rpppp(146,2)=R16_146,Rpppp(146,3)=R81_281、R81_278

Rpppp(150,2)=R16_150,Rpppp(150,3)=R81_176
Rpppp(151,2)=R16_151,Rpppp(151,3)=R81_437
Rpppp(152,2)=R16_152,Rpppp(152,3)=R81_246
Rpppp(154,2)=R16_154,Rpppp(154,3)=R81_157
Rpppp(155,2)=R16_155,Rpppp(155,3)=R81_172
Rpppp(156,2)=R16_156,Rpppp(156,3)=R81_187
Rpppp(157,2)=R16_157,Rpppp(157,3)=R81_55
Rpppp(158,2)=R16_158,Rpppp(158,3)=R81_237
Rpppp(159,2)=R16_159,Rpppp(159,3)=R81_440
Rpppp(161,2)=R16_161,Rpppp(161,3)=R81_56
Rpppp(162,2)=R16_162,Rpppp(162,3)=R81_162
Rpppp(164,2)=R16_164,Rpppp(164,3)=R81_445
Rpppp(166,2)=R16_166,Rpppp(166,3)=R81_438
Rpppp(167,2)=R16_167,Rpppp(167,3)=R81_292
Rpppp(169,2)=R16_169,Rpppp(169,3)=R81_493

Rpppp(170,2)=R16_170,Rpppp(170,3)=R81_149
Rpppp(172,2)=R16_172,Rpppp(172,3)=R81_282
Rpppp(173,2)=R16_173,Rpppp(173,3)=R81_192
Rpppp(174,2)=R16_174,Rpppp(174,3)=R81_455
Rpppp(176,2)=R16_176,Rpppp(176,3)=R81_279、R81_280
Rpppp(177,2)=R16_177,Rpppp(177,3)=R81_207
Rpppp(178,2)=R16_178,Rpppp(178,3)=R81_457
Rpppp(180,2)=R16_180,Rpppp(180,3)=R81_180

Rpppp(183,2)=R16_183,Rpppp(183,3)=R81_163
Rpppp(184,2)=R16_184,Rpppp(184,3)=R81_241
Rpppp(185,2)=R16_185,Rpppp(185,3)=R81_445
Rpppp(186,2)=R16_186,Rpppp(186,3)=R81_444
Rpppp(187,2)=R16_187,Rpppp(187,3)=R81_438
Rpppp(188,2)=R16_188,Rpppp(188,3)=R81_195
Rpppp(189,2)=R16_189,Rpppp(189,3)=R81_387

Rpppp(191,2)=R16_191,Rpppp(191,3)=R81_492
Rpppp(192,2)=R16_192,Rpppp(192,3)=R81_256
Rpppp(193,2)=R16_193,Rpppp(193,3)=R81_456
Rpppp(194,2)=R16_194,Rpppp(194,3)=R81_241
Rpppp(195,2)=R16_195,Rpppp(195,3)=R81_429
Rpppp(196,2)=R16_196,Rpppp(196,3)=R81_167
Rpppp(198,2)=R16_198,Rpppp(198,3)=R81_198

Rpppp(199,2)=R16_199,Rpppp(199,3)=R81_199
Rpppp(200,2)=R16_200,Rpppp(200,3)=R81_200
Rpppp(201,2)=R16_201,Rpppp(201,3)=R81_201
Rpppp(202,2)=R16_202,Rpppp(202,3)=R81_202
Rpppp(203,2)=R16_203,Rpppp(203,3)=R81_203
Rpppp(204,2)=R16_204,Rpppp(204,3)=R81_204
Rpppp(205,2)=R16_205,Rpppp(205,3)=R81_205
Rpppp(206,2)=R16_206,Rpppp(206,3)=R81_206
Rpppp(207,2)=R16_207,Rpppp(207,3)=R81_232
Rpppp(208,2)=R16_208,Rpppp(208,3)=R81_228
Rpppp(209,2)=R16_209,Rpppp(209,3)=R81_226
Rpppp(210,2)=R16_210,Rpppp(210,3)=R81_210
Rpppp(211,2)=R16_211,Rpppp(211,3)=R81_211
Rpppp(212,2)=R16_212,Rpppp(212,3)=R81_212
Rpppp(213,2)=R16_213,Rpppp(213,3)=R81_148
Rpppp(214,2)=R16_214,Rpppp(214,3)=R81_214
Rpppp(215,2)=R16_215,Rpppp(215,3)=R81_225
Rpppp(216,2)=R16_216,Rpppp(216,3)=R81_256
Rpppp(217,2)=R16_217,Rpppp(217,3)=R81_217
Rpppp(218,2)=R16_218,Rpppp(218,3)=R81_218
Rpppp(219,2)=R16_219,Rpppp(219,3)=R81_219
Rpppp(220,2)=R16_220,Rpppp(220,3)=R81_220
Rpppp(221,2)=R16_221,Rpppp(221,3)=R81_221
Rpppp(222,2)=R16_222,Rpppp(222,3)=R81_222
Rpppp(223,2)=R16_223,Rpppp(223,3)=R81_223
Rpppp(224,2)=R16_224,Rpppp(224,3)=R81_210
Rpppp(225,2)=R16_225,Rpppp(225,3)=R81_244
Rpppp(226,2)=R16_226,Rpppp(226,3)=R81_244
Rpppp(227,2)=R16_227,Rpppp(227,3)=R81_227
Rpppp(228,2)=R16_228,Rpppp(228,3)=R81_282
Rpppp(229,2)=R16_229,Rpppp(229,3)=R81_496
Rpppp(230,2)=R16_230,Rpppp(230,3)=R81_248
Rpppp(231,2)=R16_231,Rpppp(231,3)=R81_231
Rpppp(233,2)=R16_233,Rpppp(233,3)=R81_233、R81_462
Rpppp(235,2)=R16_235,Rpppp(235,3)=R81_254
Rpppp(236,2)=R16_236,Rpppp(236,3)=R81_458
Rpppp(237,2)=R16_237,Rpppp(237,3)=R81_255
Rpppp(238,2)=R16_238,Rpppp(238,3)=R81_238
Rpppp(239,2)=R16_239,Rpppp(239,3)=R81_169
Rpppp(240,2)=R16_240,Rpppp(240,3)=R81_240
Rpppp(242,2)=R16_242,Rpppp(242,3)=R81_242
Rpppp(243,2)=R16_243,Rpppp(243,3)=R81_243
Rpppp(245,2)=R16_245,Rpppp(245,3)=R81_245
Rpppp(247,2)=R16_247,Rpppp(247,3)=R81_247
Rpppp(249,2)=R16_249,Rpppp(249,3)=R81_249
Rpppp(250,2)=R16_250,Rpppp(250,3)=R81_250
Rpppp(251,2)=R16_251,Rpppp(251,3)=R81_251
Rpppp(252,2)=R16_252,Rpppp(252,3)=R81_252
Rpppp(254,2)=R16_254,Rpppp(254,3)=R81_166
Rpppp(255,2)=R16_255,Rpppp(255,3)=R81_229
Rpppp(256,2)=R16_256,Rpppp(256,3)=R81_235
Rpppp(257,2)=R16_257,Rpppp(257,3)=R81_257
Rpppp(258,2)=R16_258,Rpppp(258,3)=R81_258
Rpppp(259,2)=R16_259,Rpppp(259,3)=R81_259
Rpppp(260,2)=R16_260,Rpppp(260,3)=R81_260
Rpppp(261,2)=R16_261,Rpppp(261,3)=R81_439
Rpppp(262,2)=R16_262,Rpppp(262,3)=R81_436
Rpppp(263,2)=R16_263,Rpppp(263,3)=R81_263
Rpppp(264,2)=R16_264,Rpppp(264,3)=R81_264
Rpppp(265,2)=R16_265,Rpppp(265,3)=R81_265
Rpppp(266,2)=R16_266,Rpppp(266,3)=R81_266
Rpppp(268,2)=R16_268,Rpppp(268,3)=R81_268
Rpppp(269,2)=R16_269,Rpppp(269,3)=R81_269

Rpppp(271,2)=R16_271,Rpppp(271,3)=R81_261
Rpppp(272,2)=R16_272,Rpppp(272,3)=R81_262
Rpppp(273,2)=R16_273,Rpppp(273,3)=R81_417
Rpppp(274,2)=R16_274,Rpppp(274,3)=R81_317
Rpppp(275,2)=R16_275,Rpppp(275,3)=R81_316
Rpppp(276,2)=R16_276,Rpppp(276,3)=R81_391
Rpppp(277,2)=R16_277,Rpppp(277,3)=R81_413
Rpppp(278,2)=R16_278,Rpppp(278,3)=R81_409
Rpppp(279,2)=R16_279,Rpppp(279,3)=R243_0
Rpppp(280,2)=R16_280,Rpppp(280,3)=R81_424
Rpppp(281,2)=R16_281,Rpppp(281,3)=R81_410
Rpppp(282,2)=R16_282,Rpppp(282,3)=R81_392
Rpppp(283,2)=R16_283,Rpppp(283,3)=R81_407
Rpppp(284,2)=R16_284,Rpppp(284,3)=R81_380
Rpppp(285,2)=R16_285,Rpppp(285,3)=R81_343
Rpppp(286,2)=R16_286,Rpppp(286,3)=R81_414
Rpppp(287,2)=R16_287,Rpppp(287,3)=R81_389
Rpppp(288,2)=R16_288,Rpppp(288,3)=R81_386
Rpppp(289,2)=R16_289,Rpppp(289,3)=R81_318、R81_418
Rpppp(290,2)=R16_290,Rpppp(290,3)=R81_317、R81_423
Rpppp(291,2)=R16_291,Rpppp(291,3)=R81_415
Rpppp(292,2)=R16_292,Rpppp(292,3)=R81_363
Rpppp(293,2)=R16_293,Rpppp(293,3)=R81_393

Rpppp(294,2)=R16_294,Rpppp(294,3)=R81_408、R81_407
Rpppp(295,2)=R16_295,Rpppp(295,3)=R81_395
Rpppp(296,2)=R16_296,Rpppp(296,3)=R81_422
Rpppp(297,2)=R16_297,Rpppp(297,3)=R81_397
Rpppp(298,2)=R16_298,Rpppp(298,3)=R81_398
Rpppp(299,2)=R16_299,Rpppp(299,3)=R81_421
Rpppp(300,2)=R16_300,Rpppp(300,3)=R81_300
Rpppp(301,2)=R16_301,Rpppp(301,3)=R81_301
Rpppp(302,2)=R16_302,Rpppp(302,3)=R81_388
Rpppp(303,2)=R16_303,Rpppp(303,3)=R81_303
Rpppp(304,2)=R16_304,Rpppp(304,3)=R81_387
Rpppp(305,2)=R16_305,Rpppp(305,3)=R81_305
Rpppp(306,2)=R16_306,Rpppp(306,3)=R81_306
Rpppp(307,2)=R16_307,Rpppp(307,3)=R81_314
Rpppp(308,2)=R16_308,Rpppp(308,3)=R81_308
Rpppp(309,2)=R16_309,Rpppp(309,3)=R81_309
Rpppp(310,2)=R16_310,Rpppp(310,3)=R81_310
Rpppp(311,2)=R16_311,Rpppp(311,3)=R81_373
Rpppp(312,2)=R16_312,Rpppp(312,3)=R81_312
Rpppp(313,2)=R16_313,Rpppp(313,3)=R81_370
Rpppp(314,2)=R16_314,Rpppp(314,3)=R81_314
Rpppp(315,2)=R16_315,Rpppp(315,3)=R81_315
Rpppp(316,2)=R16_316,Rpppp(316,3)=R81_339
Rpppp(317,2)=R16_317,Rpppp(317,3)=R81_304
Rpppp(318,2)=R16_318,Rpppp(318,3)=R81_318
Rpppp(319,2)=R16_319,Rpppp(319,3)=R81_334
Rpppp(320,2)=R16_320,Rpppp(320,3)=R81_340
Rpppp(321,2)=R16_321,Rpppp(321,3)=R81_341
Rpppp(322,2)=R16_322,Rpppp(322,3)=R81_322
Rpppp(323,2)=R16_323,Rpppp(323,3)=R81_412
Rpppp(324,2)=R16_324,Rpppp(324,3)=R81_324
Rpppp(325,2)=R16_325,Rpppp(325,3)=R81_339
Rpppp(326,2)=R16_326,Rpppp(326,3)=R81_326
Rpppp(327,2)=R16_327,Rpppp(327,3)=R81_327
Rpppp(328,2)=R16_328,Rpppp(328,3)=R81_328
Rpppp(329,2)=R16_329,Rpppp(329,3)=R81_329
Rpppp(330,2)=R16_330,Rpppp(330,3)=R81_330
Rpppp(331,2)=R16_331,Rpppp(331,3)=R81_331
Rpppp(332,2)=R16_332,Rpppp(332,3)=R81_402
Rpppp(333,2)=R16_333,Rpppp(333,3)=R81_325
Rpppp(334,2)=R16_334,Rpppp(334,3)=R81_323
Rpppp(335,2)=R16_335,Rpppp(335,3)=R81_335
Rpppp(336,2)=R16_336,Rpppp(336,3)=R81_336
Rpppp(337,2)=R16_337,Rpppp(337,3)=R81_337
Rpppp(338,2)=R16_338,Rpppp(338,3)=R81_338
Rpppp(339,2)=R16_339,Rpppp(339,3)=R81_342
Rpppp(340,2)=R16_340,Rpppp(340,3)=R81_371
Rpppp(341,2)=R16_341,Rpppp(341,3)=R81_384
Rpppp(342,2)=R16_342,Rpppp(342,3)=R81_342
Rpppp(343,2)=R16_343,Rpppp(343,3)=R81_362
Rpppp(344,2)=R16_344,Rpppp(344,3)=R81_344
Rpppp(345,2)=R16_345,Rpppp(345,3)=R81_345
Rpppp(346,2)=R16_346,Rpppp(346,3)=R81_311
Rpppp(347,2)=R16_347,Rpppp(347,3)=R81_367
Rpppp(348,2)=R16_348,Rpppp(348,3)=R81_364
Rpppp(349,2)=R16_349,Rpppp(349,3)=R81_349
Rpppp(350,2)=R16_350,Rpppp(350,3)=R81_350
Rpppp(351,2)=R16_351,Rpppp(351,3)=R81_351
Rpppp(352,2)=R16_352,Rpppp(352,3)=R81_352
Rpppp(353,2)=R16_353,Rpppp(353,3)=R81_404
Rpppp(354,2)=R16_354,Rpppp(354,3)=R81_354
Rpppp(355,2)=R16_355,Rpppp(355,3)=R81_355
Rpppp(356,2)=R16_356,Rpppp(356,3)=R81_356
Rpppp(357,2)=R16_357,Rpppp(357,3)=R81_400
Rpppp(358,2)=R16_358,Rpppp(358,3)=R81_358
Rpppp(359,2)=R16_359,Rpppp(359,3)=R81_359
Rpppp(360,2)=R16_360,Rpppp(360,3)=R81_360
Rpppp(361,2)=R16_361,Rpppp(361,3)=R81_361
Rpppp(362,2)=R16_362,Rpppp(362,3)=R81_347
Rpppp(363,2)=R16_363,Rpppp(363,3)=R81_406
Rpppp(364,2)=R16_364,Rpppp(364,3)=R81_498
Rpppp(365,2)=R16_365,Rpppp(365,3)=R81_365
Rpppp(366,2)=R16_366,Rpppp(366,3)=R81_302
Rpppp(367,2)=R16_367,Rpppp(367,3)=R81_425
Rpppp(368,2)=R16_368,Rpppp(368,3)=R81_368
Rpppp(369,2)=R16_369,Rpppp(369,3)=R81_353
Rpppp(370,2)=R16_370,Rpppp(370,3)=R81_370
Rpppp(371,2)=R16_371,Rpppp(371,3)=R81_380
Rpppp(372,2)=R16_372,Rpppp(372,3)=R81_372
Rpppp(373,2)=R16_373,Rpppp(373,3)=R81_373
Rpppp(374,2)=R16_374,Rpppp(374,3)=R81_374、R81_405
Rpppp(375,2)=R16_375,Rpppp(375,3)=R81_375
Rpppp(376,2)=R16_376,Rpppp(376,3)=R81_376
Rpppp(377,2)=R16_377,Rpppp(377,3)=R81_420
Rpppp(378,2)=R16_378,Rpppp(378,3)=R81_378
Rpppp(379,2)=R16_379,Rpppp(379,3)=R81_379
Rpppp(380,2)=R16_380,Rpppp(380,3)=R81_380
Rpppp(381,2)=R16_381,Rpppp(381,3)=R81_381
Rpppp(382,2)=R16_382,Rpppp(382,3)=R81_394
Rpppp(383,2)=R16_383,Rpppp(383,3)=R81_383
Rpppp(384,2)=R16_384,Rpppp(384,3)=R81_332
Rpppp(385,2)=R16_385,Rpppp(385,3)=R81_385
Rpppp(387,2)=R16_387,Rpppp(387,3)=R81_388
Rpppp(388,2)=R16_388,Rpppp(388,3)=R81_388
Rpppp(389,2)=R16_389,Rpppp(389,3)=R81_321
Rpppp(390,2)=R16_390,Rpppp(390,3)=R81_390

R9_11
R:=FullMatrixAlgebra(GF(9),2);;Size(R);CR:=Center(R);;Size(CR);R:=CR;;printR(R);
R16_300:=FullMatrixAlgebra(GF(2),2);;
R81_300
R:=FullMatrixAlgebra(GF(3),2);;Size(R);printR(R);
R81_321
R1:=PolynomialRing(GF(9),1);;x:=Indeterminate(GF(9));;I2:=IdealByGenerators(R1,[x^2+x]);;R:=R1/I2;;printR(R);
R81_383
R1:=PolynomialRing(GF(9),1);;x:=Indeterminate(GF(9));;I1:=IdealByGenerators(R1,[x^2]);;R:=R1/I1;;printR(R);
R16_6:=DirectSum(ZmodnZ(8),ZmodnZ(2));;
R81_6
R:=DirectSum(ZmodnZ(27),ZmodnZ(3));;Size(R);printR(R);
R16_100:=DirectSum(ZmodnZ(4),ZmodnZ(4));;
R81_104
R:=DirectSum(ZmodnZ(9),ZmodnZ(9));;Size(R);printR(R);
R16_200:=DirectSum(DirectSum(GF(2),GF(2)),ZmodnZ(4));;
R81_244
R:=DirectSum(DirectSum(GF(3),GF(3)),ZmodnZ(9));;Size(R);printR(R);

R81_98
m:=27;;I:=[ [ ZmodnZObj( 3, m), ZmodnZObj( 0, m) ], [ ZmodnZObj( 0, m), ZmodnZObj( 3, m) ] ];;A2:=[ [ ZmodnZObj( 9, m), ZmodnZObj( 0, m) ], [ ZmodnZObj( 0, m), ZmodnZObj( 3, m) ] ];;R:=RingByGenerators([I,A2]);;n:=Size(R);printR(R);

R27_49
F4:=Elements(GF(9));A:=[[F4[1],F4[3]], [F4[1],F4[3]]];B:=[[F4[3],F4[3]], [F4[1],F4[1]]];;R:=RingByGenerators([A,B]);;Size(R);printR(R);
81阶全矩阵环M_2(M_3)=R81_301
R2:=Elements(SmallRing(3,1));A:=[[R2[1],R2[2]], [R2[2],R2[1]]];;B:=[[R2[2],R2[1]], [R2[1],R2[2]]];;C:=[[R2[1],R2[1]], [R2[2],R2[1]]];;D:=[[R2[2],R2[1]], [R2[2],R2[1]]];;R:=RingByGenerators([A,B,C,D]);;Size(R);printR(R);
81阶全矩阵环M_2(F_3)=R81_300
R2:=Elements(SmallRing(3,2));A:=[[R2[1],R2[2]], [R2[2],R2[1]]];;B:=[[R2[2],R2[1]], [R2[1],R2[2]]];;C:=[[R2[1],R2[1]], [R2[2],R2[1]]];;D:=[[R2[2],R2[1]], [R2[2],R2[1]]];;R:=RingByGenerators([A,B,C,D]);;Size(R);printR(R);
R81_321
F4:=Elements(GF(9));A:=[[F4[2],F4[4]], [F4[4],F4[2]]];B:=[[F4[1],F4[3]], [F4[3],F4[1]]];;R:=RingByGenerators([A,B]);;Size(R);printR(R);

vector<int> IdealProduct(IRing* r,const vector<int>& A,const vector<int>& B){
    int nA=A.size();	
    int nB=B.size();
    set<int> S;	
	for (int i=0; i<nA; i++){
		int I=0;
		for (int j=0; j<nB; j++){
			int ij=r->mul(A[i],B[j]);
			I=r->add(I,ij);
			S.insert(ij);
			S.insert(I);
		}
		S.insert(I);
	}
	vector<int> V;
	V.assign(S.begin(),S.end());	
	int cnt=0,cnt1=0;
	do{
		cnt=V.size();
		if(cnt==1)
			break;
		for(int i=0;i<cnt;i++){
			for(int j=0;j<cnt;j++){
				int IJ=r->add(V[i],V[j]);
				vector<int>::iterator p=std::find(V.begin(),V.end(),IJ);
				if(p==V.end()){
					V.push_back(IJ);
				}
			}
		}
		cnt1=V.size();
	}while(cnt1!=cnt);
	if(V.size()>1){
		std::sort(V.begin(),V.end());
	}
	return V;
}

//>0表示幂零环的幂零指数，0表示非幂零环
int IsNilpotent(IRing* r){
	int n=r->size();
	vector<int> S0;
	for(int i=0;i<n;i++){
		S0.push_back(i);
	}
	vector<int> S=S0;	
	int cnt=S.size();	
	if(cnt==1)
		return 1;
	int iret=1;	
	bool bE=false;
	do{
		vector<int> S1=IdealProduct(r,S,S0);
		iret++;		
		bE=IsEqual(S1,S);
		S=S1;
		if(S.size()==1)
			return iret; 	
	}while(!bE);
	//string str=V2S(S);
	//printf("iret=%d,cnt=%d,str=%s\n",iret,S.size(),str.c_str());
    return 0;
}

E:\MathTool\gaptool>for /L %x in (1,1,34) do bN R16%x.txt

E:\MathTool\gaptool>bN R161.txt
1,2

E:\MathTool\gaptool>bN R162.txt
2,5

E:\MathTool\gaptool>bN R163.txt
3,0

E:\MathTool\gaptool>bN R164.txt
4,3

E:\MathTool\gaptool>bN R165.txt
5,3

E:\MathTool\gaptool>bN R166.txt
6,0

E:\MathTool\gaptool>bN R167.txt
7,0

E:\MathTool\gaptool>bN R168.txt
8,4

E:\MathTool\gaptool>bN R169.txt
9,3

E:\MathTool\gaptool>bN R1610.txt
10,4

E:\MathTool\gaptool>bN R1611.txt
11,0

E:\MathTool\gaptool>bN R1612.txt
12,2

E:\MathTool\gaptool>bN R1613.txt
13,0

E:\MathTool\gaptool>bN R1614.txt
14,0

E:\MathTool\gaptool>bN R1615.txt
15,0

E:\MathTool\gaptool>bN R1616.txt
16,3

E:\MathTool\gaptool>bN R1617.txt
17,4

E:\MathTool\gaptool>bN R1618.txt
18,3

E:\MathTool\gaptool>bN R1619.txt
19,4

E:\MathTool\gaptool>bN R1620.txt
20,0

E:\MathTool\gaptool>bN R1621.txt
21,3

E:\MathTool\gaptool>bN R1622.txt
22,3

E:\MathTool\gaptool>bN R1623.txt
23,0

E:\MathTool\gaptool>bN R1624.txt
24,4

E:\MathTool\gaptool>bN R1625.txt
25,4

E:\MathTool\gaptool>bN R1626.txt
26,3

E:\MathTool\gaptool>bN R1627.txt
27,3

E:\MathTool\gaptool>bN R1628.txt
28,3

E:\MathTool\gaptool>bN R1629.txt
29,3

E:\MathTool\gaptool>bN R1630.txt
30,4

E:\MathTool\gaptool>bN R1631.txt
31,3

E:\MathTool\gaptool>bN R1632.txt
32,0

E:\MathTool\gaptool>bN R1633.txt
33,3

E:\MathTool\gaptool>bN R1634.txt
34,4

E:\MathTool\gaptool>