p^3阶有限环序列以及R27的编号

gap> calT(27);
4
1~4：G27_1=C27,N0=1,2,6,18,
5~24、32、53~55：G27_2=C9 x C3,N0=1,8,18,0,
25~52（缺40）、56~59：G27_5=C3 x C3 x C3,N0=1,26,0,0,

D:\MathTool\gaptool>R27
8阶交换环已找到34种(总共34种):
1,2,3,4,5,6,7,11,12,13,14,16,17,19,21,22,23,24,25,26,27,31,32,33,35,38,40,42,45,46,48,50,51,52,
8阶非交换环已找到18种(总共18种):
8,9,10,15,18,20,28,29,30,34,36,37,39,41,43,44,47,49,
8阶幺环已找到11种(总共11种):
3,14,21,22,45,46,48,49,50,51,52,

27阶交换环已找到35种(总共36种):
1,2,3,4,5,7,9,11,12,13,14,16,17,21,22,23,24,25,27,29,31,32,33,35,38,42,45,46,50,51,52,54,56,57,59,
27阶非交换环已找到23种(总共23种):
6,8,10,15,18,19,20,26,28,30,34,36,37,39,41,43,44,47,48,49,53,55,58,
27阶幺环已找到12种(总共12种):
3,14,21,22,32,45,46,49,50,51,52,59,

D:\MathTool\gaptool>Rpppp 1 52 8
ret=0,环表示数据表中的记录条数=853
ret2=0,环表示数据表中的记录条数=893
Rpppp(1,2)=R8_1,Rpppp(1,3)=R27_4
Rpppp(2,2)=R8_2,Rpppp(2,3)=R27_3
Rpppp(3,2)=R8_3,Rpppp(3,3)=R27_3
Rpppp(4,2)=R8_4,Rpppp(4,3)=R27_4
Rpppp(5,2)=R8_5,Rpppp(5,3)=R27_5
Rpppp(7,2)=R8_7,Rpppp(7,3)=R27_7
Rpppp(8,2)=R8_8,Rpppp(8,3)=R27_8
Rpppp(10,2)=R8_10,Rpppp(10,3)=R27_10
Rpppp(11,2)=R8_11,Rpppp(11,3)=R27_11
Rpppp(13,2)=R8_13,Rpppp(13,3)=R27_13
Rpppp(15,2)=R8_15,Rpppp(15,3)=R27_15
Rpppp(16,2)=R8_16,Rpppp(16,3)=R27_16
Rpppp(17,2)=R8_17,Rpppp(17,3)=R27_14
Rpppp(19,2)=R8_19,Rpppp(19,3)=R27_19
Rpppp(20,2)=R8_20,Rpppp(20,3)=R27_20
Rpppp(21,2)=R8_21,Rpppp(21,3)=R27_21
Rpppp(23,2)=R8_23,Rpppp(23,3)=R27_23
Rpppp(24,2)=R8_24,Rpppp(24,3)=R27_24
Rpppp(25,2)=R8_25,Rpppp(25,3)=R27_25
Rpppp(26,2)=R8_26,Rpppp(26,3)=R27_26
Rpppp(27,2)=R8_27,Rpppp(27,3)=R27_27
Rpppp(28,2)=R8_28,Rpppp(28,3)=R27_28
Rpppp(29,2)=R8_29,Rpppp(29,3)=R81_300
Rpppp(30,2)=R8_30,Rpppp(30,3)=R27_30
Rpppp(31,2)=R8_31,Rpppp(31,3)=R27_31
Rpppp(32,2)=R8_32,Rpppp(32,3)=R27_29
Rpppp(33,2)=R8_33,Rpppp(33,3)=R27_33
Rpppp(34,2)=R8_34,Rpppp(34,3)=R81_300
Rpppp(35,2)=R8_35,Rpppp(35,3)=R27_35
Rpppp(36,2)=R8_36,Rpppp(36,3)=R27_36
Rpppp(37,2)=R8_37,Rpppp(37,3)=R27_37
Rpppp(38,2)=R8_38,Rpppp(38,3)=R27_38
Rpppp(39,2)=R8_39,Rpppp(39,3)=R27_39
Rpppp(40,2)=R8_40,Rpppp(40,3)=R27_38
Rpppp(41,2)=R8_41,Rpppp(41,3)=R27_41
Rpppp(42,2)=R8_42,Rpppp(42,3)=R27_42
Rpppp(43,2)=R8_43,Rpppp(43,3)=R27_43
Rpppp(44,2)=R8_44,Rpppp(44,3)=R27_44
Rpppp(45,2)=R8_45,Rpppp(45,3)=R27_45
Rpppp(46,2)=R8_46,Rpppp(46,3)=R27_46
Rpppp(47,2)=R8_47,Rpppp(47,3)=R27_47
Rpppp(48,2)=R8_48,Rpppp(48,3)=R27_50
Rpppp(49,2)=R8_49,Rpppp(49,3)=R27_49
Rpppp(50,2)=R8_50,Rpppp(50,3)=R27_50
Rpppp(51,2)=R8_51,Rpppp(51,3)=R27_51
Rpppp(52,2)=R8_52,Rpppp(52,3)=R27_52

D:\MathTool\gaptool>Rppp 1 59
Rppp(1,2)=R8_1,Rppp(1,3)=R27_1
Rppp(2,2)=R8_2,Rppp(2,3)=R27_2
Rppp(3,2)=R8_3,Rppp(3,3)=R27_3
Rppp(4,2)=R8_4,Rppp(4,3)=R27_4
Rppp(5,2)=R8_5,Rppp(5,3)=R27_5
Rppp(7,2)=R8_7,Rppp(7,3)=R27_7
Rppp(8,2)=R8_8,Rppp(8,3)=R27_8
Rppp(10,2)=R8_10,Rppp(10,3)=R27_10
Rppp(11,2)=R8_11,Rppp(11,3)=R27_11
Rppp(13,2)=R8_13,Rppp(13,3)=R27_13
Rppp(14,2)=R8_14,Rppp(14,3)=R27_14
Rppp(15,2)=R8_15,Rppp(15,3)=R27_15
Rppp(16,2)=R8_16,Rppp(16,3)=R27_16
Rppp(17,2)=R8_17,Rppp(17,3)=R27_17
Rppp(19,2)=R8_19,Rppp(19,3)=R27_19
Rppp(20,2)=R8_20,Rppp(20,3)=R27_20
Rppp(21,2)=R8_21,Rppp(21,3)=R27_21
Rppp(23,2)=R8_23,Rppp(23,3)=R27_23
Rppp(24,2)=R8_24,Rppp(24,3)=R27_24
Rppp(25,2)=R8_25,Rppp(25,3)=R27_25
Rppp(26,2)=R8_26,Rppp(26,3)=R27_26
Rppp(27,2)=R8_27,Rppp(27,3)=R27_27
Rppp(28,2)=R8_28,Rppp(28,3)=R27_28
Rppp(29,2)=R8_29,Rppp(29,3)=R27_49
Rppp(30,2)=R8_30,Rppp(30,3)=R27_30
Rppp(31,2)=R8_31,Rppp(31,3)=R27_31
Rppp(32,2)=R8_32,Rppp(32,3)=R27_29
Rppp(33,2)=R8_33,Rppp(33,3)=R27_33
Rppp(34,2)=R8_34,Rppp(34,3)=R27_49
Rppp(35,2)=R8_35,Rppp(35,3)=R27_35
Rppp(36,2)=R8_36,Rppp(36,3)=R27_36
Rppp(37,2)=R8_37,Rppp(37,3)=R27_37
Rppp(38,2)=R8_38,Rppp(38,3)=R27_38
Rppp(39,2)=R8_39,Rppp(39,3)=R27_39
Rppp(40,2)=R8_40,Rppp(40,3)=R27_38
Rppp(41,2)=R8_41,Rppp(41,3)=R27_41
Rppp(42,2)=R8_42,Rppp(42,3)=R27_42
Rppp(43,2)=R8_43,Rppp(43,3)=R27_43
Rppp(44,2)=R8_44,Rppp(44,3)=R27_44
Rppp(45,2)=R8_45,Rppp(45,3)=R27_45
Rppp(46,2)=R8_46,Rppp(46,3)=R27_46
Rppp(47,2)=R8_47,Rppp(47,3)=R27_47
Rppp(48,2)=R8_48,Rppp(48,3)=R27_46
Rppp(49,2)=R8_49,Rppp(49,3)=R27_49
Rppp(50,2)=R8_50,Rppp(50,3)=R27_50
Rppp(51,2)=R8_51,Rppp(51,3)=R27_51
Rppp(52,2)=R8_52,Rppp(52,3)=R27_52
Rppp(56,2)=R8_26,Rppp(56,3)=R27_56
Rppp(57,2)=R8_33,Rppp(57,3)=R27_57
Rppp(59,2)=R8_52,Rppp(59,3)=R27_59

for j in [1..2] do for i in [1..11] do R:=DirectSum(SmallRing(9,i),SmallRing(3,j));;ni:=(j-1)*11+i;Print("R9R3_",ni,":");printR(R);od;od;
R9R3_1=R9_1×R3_1=R27_5
R9R3_2=R9_2×R3_1=R27_13
R9R3_3=R9_3×R3_1=R27_16
R9R3_4=R9_4×R3_1=R27_25
R9R3_5=R9_5×R3_1=R27_56
R9R3_6=R9_6×R3_1=R27_27
R9R3_7=R9_7×R3_1=R27_26
R9R3_8=R9_8×R3_1=R27_28
R9R3_9=R9_9×R3_1=R27_35
R9R3_10=R9_10×R3_1=R27_38
R9R3_11=R9_11×R3_1=R27_42
R9R3_12=R9_1×R3_2=R27_7
R9R3_13=R9_2×R3_2=R27_14
R9R3_14=R9_3×R3_2=R27_17
R9R3_15=R9_4×R3_2=R27_27
R9R3_16=R9_5×R3_2=R27_33
R9R3_17=R9_6×R3_2=R27_38
R9R3_18=R9_7×R3_2=R27_41
R9R3_19=R9_8×R3_2=R27_47
R9R3_20=R9_9×R3_2=R27_46
R9R3_21=R9_10×R3_2=R27_50
R9R3_22=R9_11×R3_2=R27_51

R9_1
m:=3;;R:=RingByGenerators([ZmodnZObj(m*m,m*m*m*m)]);;printR(R);
R9_2
m:=3;;R:=ZmodnZ(m*m);;printR(R);
R9_3
m:=3;;R:=RingByGenerators([ZmodnZObj(m,m*m*m)]);;printR(R);
R9_4
m:=3;;a:=[ [ ZmodnZObj(m,m*m), ZmodnZObj( 0, m*m) ], [ ZmodnZObj( 0, m*m), ZmodnZObj( 0, m*m) ] ];;b:=[ [ ZmodnZObj(0,m*m), ZmodnZObj( 0, m*m) ], [ ZmodnZObj( 0, m*m), ZmodnZObj( m, m*m) ] ];;R:=RingByGenerators([a,b]);;printR(R);
m:=3;;R:=DirectSum(SmallRing(m,1),SmallRing(m,1));;printR(R);
R9_5
m:=3;;A:=[ [ ZmodnZObj( 0, m), ZmodnZObj( 0, m), ZmodnZObj( 0, m) ], [ ZmodnZObj( 0, m), ZmodnZObj( 0, m), ZmodnZObj( 1, m) ], [ ZmodnZObj( 1, m), ZmodnZObj( 0, m), ZmodnZObj( 0, m) ] ];;R:=RingByGenerators([A]);;printR(R);
R9_6
m:=3;;S:=RingByGenerators([DirectProductElement( [ 0*Z(m), ZmodnZObj(m,m*m)]),DirectProductElement( [ Z(m)^0, ZmodnZObj(m,m*m)])]);;Size(S);printR(S);
m:=3;;R:=DirectSum(SmallRing(m,1),SmallRing(m,2));;printR(R);
R9_7
m:=3;;I:=[ [ ZmodnZObj( 1, m), ZmodnZObj( 0, m) ], [ ZmodnZObj( 0, m), ZmodnZObj( 1, m) ] ];;A:=[ [ ZmodnZObj( 1, m), ZmodnZObj( 0, m) ], [ ZmodnZObj( 0, m), ZmodnZObj( 0, m) ] ];;B:=[ [ ZmodnZObj( 1, m), ZmodnZObj( 1, m) ], [ ZmodnZObj( 0, m), ZmodnZObj( 1, m) ] ];;C:=[ [ ZmodnZObj( 1, m), ZmodnZObj( 1, m) ], [ ZmodnZObj( 0, m), ZmodnZObj( 0, m) ] ];;R:=RingByGenerators([A,C]);;printR(R);
R9_8
m:=3;;I:=[ [ ZmodnZObj( 1, m), ZmodnZObj( 0, m) ], [ ZmodnZObj( 0, m), ZmodnZObj( 1, m) ] ];;A:=[ [ ZmodnZObj( 1, m), ZmodnZObj( 0, m) ], [ ZmodnZObj( 0, m), ZmodnZObj( 0, m) ] ];;B:=[ [ ZmodnZObj( 1, m), ZmodnZObj( 1, m) ], [ ZmodnZObj( 0, m), ZmodnZObj( 1, m) ] ];;C:=[ [ ZmodnZObj( 1, m), ZmodnZObj( 1, m) ], [ ZmodnZObj( 0, m), ZmodnZObj( 0, m) ] ];;D:=[ [ ZmodnZObj( 1, m), ZmodnZObj( 0, m) ], [ ZmodnZObj( 1, m), ZmodnZObj( 0, m) ] ];;R:=RingByGenerators([A,D]);;printR(R);
R9_9
m:=3;;R1:=PolynomialRing(GF(m),1);;x:=Indeterminate(GF(m));;poly:=x^2+x+1;;IsIrreducibleRingElement(R1,poly);Degree(poly);;Factors(poly);;I:=IdealByGenerators(R1,[poly]);;R:=R1/I;;printR(R);
m:=3;;R1:=PolynomialRing(GF(m),1);;x:=Indeterminate(GF(m));;poly:=x^2;;IsIrreducibleRingElement(R1,poly);Degree(poly);;Factors(poly);;I:=IdealByGenerators(R1,[poly]);;R:=R1/I;;printR(R);
R9_10
m:=3;;R1:=PolynomialRing(GF(m),1);;x:=Indeterminate(GF(m));;poly:=x^2+x;;IsIrreducibleRingElement(R1,poly);Degree(poly);;Factors(poly);;I:=IdealByGenerators(R1,[poly]);;R:=R1/I;;printR(R);
m:=3;;R:=DirectSum(SmallRing(m,2),SmallRing(m,2));;printR(R);
R9_11
m:=3;;R1:=PolynomialRing(GF(m),1);;x:=Indeterminate(GF(m));;poly:=x^2+1;;IsIrreducibleRingElement(R1,poly);Degree(poly);;Factors(poly);;I:=IdealByGenerators(R1,[poly]);;R:=R1/I;;printR(R);

R27_1
R:=RingByGenerators([ZmodnZObj(27,729)]);;printR(R);
R27_2
R:=RingByGenerators([ZmodnZObj(3,81)]);;printR(R);
R27_3
R:=RingByGenerators([ZmodnZObj(1,27)]);;printR(R);
R27_4
R:=RingByGenerators([ZmodnZObj(9,243)]);;printR(R);
R27_49
m:=3;;A:=[ [ ZmodnZObj( 1, m), ZmodnZObj( 0, m) ], [ ZmodnZObj( 0, m), ZmodnZObj( 0, m) ] ];;B:=[ [ ZmodnZObj( 1, m), ZmodnZObj( 1, m) ], [ ZmodnZObj( 0, m), ZmodnZObj( 1, m) ] ];;R:=RingByGenerators([A,B]);;printR(R);

R27_59
R1:=PolynomialRing(GF(3),1);;x:=Indeterminate(GF(3));;poly:=x^3+2*x+2;;IsIrreducibleRingElement(R1,poly);Degree(poly);;Factors(poly);;I:=IdealByGenerators(R1,[poly]);;R:=R1/I;;printR(R);
true