11种4阶环、22种12阶环

只讨论4阶环、12阶环等阶数的有限环，其他阶数的有限环不在这篇文章里讨论。

R4_3=ZimodnZObj(1,4)=Z/(3)
R4_4={0,2,2i,2+2i}mod4
R4_5=ZimodnZObj(1+i,2+2i)

Z[i]=Z[x]/(x^2+1)
[1,0,1]在Z/(2)上有根，R4_9=Z[i]/(2)=Z/(2)[x]/(x^2+1)=F_2[u]/(u^2+1)=F_2[u]/(u^2),(Z[i]/(2))^*=C_2
0->[0]
1->[1]
2->[0,1]
3->[1,1]
[0,1,1]在Z/(2)上有根，R4_10=F_2[v]/(v(v+1))
[1,1,1]在Z/(2)上无根，R4_11=GF(2,2)=Z[ω]/(2)=Z/(2)[x]/(x^2+x+1):
0->[0]
1->[1]
2->[0,1]
3->[1,1]

20151107：对于4阶环的分类，可以减去1个环不变量n1。GAP4中Characteristic命令得到的有限环的特征不一定准确，可以根据环的加法群的凯莱表计算得到这个环不变量。
gap> m:=4;;n:=NumberSmallRings(m);;for ni in [1..n] do R:=SmallRing(m,ni);;L:=Elements(R);;n1:=0;;for i1 in L do if InverseMutable(i1)=fail then n1:=n1+1;fi;od;n2:=0;;for i2 in L do if IsIdempotent(i2) then n2:=n2+1;fi;od;n3:=0;;for i3 in L do if IsOne(i3) then n3:=n3+1;fi;od;n4:=0;;for i4 in L do if IsZero(i4)=false and i4^2=Zero(R) then n4:=n4+1;fi;od;n5:=0;;for i5 in L do if IsZero(i5)=false and i5^3=Zero(R) then n5:=n5+1;fi;od;n6:=0;;for i6 in L do for j6 in L do if IsZero(i6*j6) then n6:=n6+1;fi;od;od;bA:=0;bO:=0;if IsAbelian(R) then bA:=1;fi;if n3=1 then bO:=1;fi;Print("环R",m,"_",ni,"的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=",Characteristic(R),",",bA,",",bO,",",n1,",",n2,",",n4,",",n5,",",n6,",");n7:=0;;for i7 in L do for j7 in L do if IsZero(i7)=false and IsZero(j7)=false and IsZero(i7*j7) then n7:=n7+1;break;fi;od;od;CR:=Center(R);;n8:=Size(CR);M:=Ideals(R);;LL:=[];;for i in [1..m] do Add(LL,Size(RingByGenerators([L[i]])));od;Sort(LL);Print(n7,",",n8,",",List(M,Size),",",List(LL),"\n");od;
环R4_1的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=4,1,0,4,1,3,3,16,3,4,[ 1, 2, 4 ],[ 1, 2, 4, 4 ]
环R4_2的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=4,1,0,4,1,1,3,12,3,4,[ 1, 2, 4 ],[ 1, 2, 4, 4 ]
环R4_3的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=4,1,1,2,2,1,1,8,1,4,[ 1, 2, 4 ],[ 1, 2, 4, 4 ]
环R4_4的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=2,1,0,4,1,3,3,16,3,4,[ 1, 2, 2, 2, 4 ],[ 1, 2, 2, 2 ]
环R4_5的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=2,1,0,4,1,1,3,12,3,4,[ 1, 2, 4 ],[ 1, 2, 4, 4 ]
环R4_6的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=2,1,0,4,2,1,1,12,3,4,[ 1, 2, 2, 4 ],[ 1, 2, 2, 4 ]
环R4_7的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=2,0,0,4,3,1,1,10,1,1,[ 1, 2, 4 ],[ 1, 2, 2, 2 ]
环R4_8的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=2,0,0,4,3,1,1,10,3,1,[ 1, 2, 4 ],[ 1, 2, 2, 2 ]
环R4_9的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=2,1,1,2,2,1,1,8,1,4,[ 1, 2, 4 ],[ 1, 2, 2, 4 ]
环R4_10的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=2,1,1,3,4,0,0,9,2,4,[ 1, 2, 2, 4 ],[ 1, 2, 2, 2 ]
环R4_11的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=2,1,1,1,2,0,0,7,0,4,[ 1, 4 ],[ 1, 2, 4, 4 ]
gap> R4_1:=RingByGenerators([ZmodnZObj(4,16)]);;R4_2:=RingByGenerators([ZmodnZObj(2,8)]);;R4_3:=RingByGenerators([ZmodnZObj(3,12)]);;m:=4;;a:=[ [ ZmodnZObj(2,m), ZmodnZObj( 0, m) ], [ ZmodnZObj( 0, m), ZmodnZObj( 0, m) ] ];;b:=[ [ ZmodnZObj(0,m), ZmodnZObj( 0, m) ], [ ZmodnZObj( 0, m), ZmodnZObj( 2, m) ] ];;R4_4:=RingByGenerators([a,b]);;R4_4a:=DirectSum(SmallRing(2,1),SmallRing(2,1));;R4_6:=RingByGenerators([DirectProductElement( [ 0*Z(2), ZmodnZObj(2,4)]),DirectProductElement( [ Z(2)^0, ZmodnZObj(2,4)])]);;m:=2;;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) ] ];;R4_7:=RingByGenerators([A,C]);;m:=2;;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) ] ];;R4_8:=RingByGenerators([A,D]);;m:=2;;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) ] ];;R4_9:=RingByGenerators([I,B]);;I:=[ [ ZmodnZObj( 1, 2 ), ZmodnZObj( 0, 2 ) ], [ ZmodnZObj( 0, 2 ), ZmodnZObj( 1, 2 ) ] ];;A:=[ [ ZmodnZObj( 1, 2 ), ZmodnZObj( 0, 2 ) ], [ ZmodnZObj( 0, 2 ), ZmodnZObj( 0, 2 ) ] ];;R4_10:=RingByGenerators([I,A]);;O:=[[0*Z(2),0*Z(2)],[0*Z(2),0*Z(2)]];;I:=[[Z(2),0*Z(2)],[0*Z(2),Z(2)]];;B:=[[Z(2),Z(2)],[Z(2),0*Z(2)]];;A:=[[0*Z(2),Z(2)],[Z(2),Z(2)]];;K4:=GroupWithGenerators([I,A,B]);;IdGroup(K4);;R4_11:=RingByGenerators([O,I,A,B]);;R4:=[R4_1,R4_2,R4_3,R4_4,R4_4a,R4_6,R4_7,R4_8,R4_9,R4_10,R4_11];;for R in R4 do m:=4;;ni:=Position(R4,R);;L:=Elements(R);;n1:=0;;for i1 in L do if InverseMutable(i1)=fail then n1:=n1+1;fi;od;n2:=0;;for i2 in L do if IsIdempotent(i2) then n2:=n2+1;fi;od;n3:=0;;for i3 in L do if IsOne(i3) then n3:=n3+1;fi;od;n4:=0;;for i4 in L do if IsZero(i4)=false and i4^2=Zero(R) then n4:=n4+1;fi;od;n5:=0;;for i5 in L do if IsZero(i5)=false and i5^3=Zero(R) then n5:=n5+1;fi;od;n6:=0;;for i6 in L do for j6 in L do if IsZero(i6*j6) then n6:=n6+1;fi;od;od;bA:=0;;bO:=0;;if IsAbelian(R) then bA:=1;fi;if n3=1 then bO:=1;fi;n7:=0;;for i7 in L do for j7 in L do if IsZero(i7)=false and IsZero(j7)=false and IsZero(i7*j7) then n7:=n7+1;break;fi;od;od;CR:=Center(R);;n8:=Size(CR);;Print("环R",m,"_",ni,"的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=",Characteristic(R),",",bA,",",bO,",",n1,",",n2,",",n4,",",n5,",",n6,",",n7,",",n8,"\n");od;
环R4_1的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=16,1,0,4,1,3,3,16,3,4
环R4_2的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=8,1,0,4,1,1,3,12,3,4
环R4_3的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=12,1,0,4,2,1,1,8,1,4
环R4_4的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=4,1,0,4,1,3,3,16,3,4
环R4_4a的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=2,1,0,4,1,3,3,16,3,4
环R4_6的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=fail,1,0,0,2,1,1,12,3,4
环R4_7的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=2,0,0,4,3,1,1,10,1,1
环R4_8的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=2,0,0,4,3,1,1,10,3,1
环R4_9的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=2,1,1,2,2,1,1,8,1,4
环R4_10的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=2,1,1,3,4,0,0,9,2,4
环R4_11的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=2,1,1,1,2,0,0,7,0,4
环R8_38的两种4阶子环R4_5、R4_6：
gap> m:=2;;I3:=[[0*Z(m),Z(m),0*Z(m),Z(m)],[Z(m),Z(m),0*Z(m),0*Z(m)],[0*Z(m),0*Z(m),Z(m),Z(m)],[0*Z(m),Z(m),0*Z(m),Z(m)]];;R8_38:=RingByGenerators([I3]);;m:=8;;ni:=38;;R:=R8_38;;L:=Elements(R);;n1:="未知";;n2:=0;;for i2 in L do if IsIdempotent(i2) then n2:=n2+1;fi;od;n3:=0;;for i3 in L do if IsOne(i3) then n3:=n3+1;fi;od;n4:=0;;for i4 in L do if IsZero(i4)=false and i4^2=Zero(R) then n4:=n4+1;fi;od;n5:=0;;for i5 in L do if IsZero(i5)=false and i5^3=Zero(R) then n5:=n5+1;fi;od;n6:=0;;for i6 in L do for j6 in L do if IsZero(i6*j6) then n6:=n6+1;fi;od;od;bA:=0;bO:=0;if IsAbelian(R) then bA:=1;fi;if n3=1 then bO:=1;fi;n7:=0;;for i7 in L do for j7 in L do if IsZero(i7)=false and IsZero(j7)=false and IsZero(i7*j7) then n7:=n7+1;break;fi;od;od;CR:=Center(R);;n8:=Size(CR);LL:=[];;for i in [1..m] do Add(LL,Size(RingByGenerators([L[i]])));od;Sort(LL);Print("环R",m,"_",ni,"的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=",Characteristic(R),",",bA,",",bO,",",n1,",",n2,",",n4,",",n5,",",n6,",",n7,",",n8,",",List(LL),"\n");     
环R8_38的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=2,1,0,未知,2,1,3,36,7,8,[ 1, 2, 2, 4, 4, 4, 8, 8 ]
gap> m:=2;;I3:=[[0*Z(m),Z(m),0*Z(m),Z(m)],[Z(m),Z(m),0*Z(m),0*Z(m)],[0*Z(m),0*Z(m),Z(m),Z(m)],[0*Z(m),Z(m),0*Z(m),Z(m)]];;R4_5:=RingByGenerators([I3*I3+I3]);;m:=4;;ni:=5;;R:=R4_5;;L:=Elements(R);;n1:="未知";;n2:=0;;for i2 in L do if IsIdempotent(i2) then n2:=n2+1;fi;od;n3:=0;;for i3 in L do if IsOne(i3) then n3:=n3+1;fi;od;n4:=0;;for i4 in L do if IsZero(i4)=false and i4^2=Zero(R) then n4:=n4+1;fi;od;n5:=0;;for i5 in L do if IsZero(i5)=false and i5^3=Zero(R) then n5:=n5+1;fi;od;n6:=0;;for i6 in L do for j6 in L do if IsZero(i6*j6) then n6:=n6+1;fi;od;od;bA:=0;bO:=0;if IsAbelian(R) then bA:=1;fi;if n3=1 then bO:=1;fi;n7:=0;;for i7 in L do for j7 in L do if IsZero(i7)=false and IsZero(j7)=false and IsZero(i7*j7) then n7:=n7+1;break;fi;od;od;CR:=Center(R);;n8:=Size(CR);LL:=[];;for i in [1..m] do Add(LL,Size(RingByGenerators([L[i]])));od;Sort(LL);Print("环R",m,"_",ni,"的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=",Characteristic(R),",",bA,",",bO,",",n1,",",n2,",",n4,",",n5,",",n6,",",n7,",",n8,",",List(LL),"\n");
环R4_5的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=2,1,0,未知,1,1,3,12,3,4,[ 1, 2, 4, 4 ]
gap> m:=2;;I3:=[[0*Z(m),Z(m),0*Z(m),Z(m)],[Z(m),Z(m),0*Z(m),0*Z(m)],[0*Z(m),0*Z(m),Z(m),Z(m)],[0*Z(m),Z(m),0*Z(m),Z(m)]];;R4_6:=RingByGenerators([I3*I3]);;m:=4;;ni:=6;;R:=R4_6;;L:=Elements(R);;n1:="未知";;n2:=0;;for i2 in L do if IsIdempotent(i2) then n2:=n2+1;fi;od;n3:=0;;for i3 in L do if IsOne(i3) then n3:=n3+1;fi;od;n4:=0;;for i4 in L do if IsZero(i4)=false and i4^2=Zero(R) then n4:=n4+1;fi;od;n5:=0;;for i5 in L do if IsZero(i5)=false and i5^3=Zero(R) then n5:=n5+1;fi;od;n6:=0;;for i6 in L do for j6 in L do if IsZero(i6*j6) then n6:=n6+1;fi;od;od;bA:=0;bO:=0;if IsAbelian(R) then bA:=1;fi;if n3=1 then bO:=1;fi;n7:=0;;for i7 in L do for j7 in L do if IsZero(i7)=false and IsZero(j7)=false and IsZero(i7*j7) then n7:=n7+1;break;fi;od;od;CR:=Center(R);;n8:=Size(CR);LL:=[];;for i in [1..m] do Add(LL,Size(RingByGenerators([L[i]])));od;Sort(LL);Print("环R",m,"_",ni,"的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=",Characteristic(R),",",bA,",",bO,",",n1,",",n2,",",n4,",",n5,",",n6,",",n7,",",n8,",",List(LL),"\n");
环R4_6的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=2,1,0,未知,2,1,1,12,3,4,[ 1, 2, 2, 4 ]
gap> m:=12;;n:=NumberSmallRings(m);;for ni in [1..n] do R:=SmallRing(m,ni);;L:=Elements(R);;n1:=0;;for i1 in L do if InverseMutable(i1)=fail then n1:=n1+1;fi;od;n2:=0;;for i2 in L do if IsIdempotent(i2) then n2:=n2+1;fi;od;n3:=0;;for i3 in L do if IsOne(i3) then n3:=n3+1;fi;od;n4:=0;;for i4 in L do if IsZero(i4)=false and i4^2=Zero(R) then n4:=n4+1;fi;od;n5:=0;;for i5 in L do if IsZero(i5)=false and i5^3=Zero(R) then n5:=n5+1;fi;od;n6:=0;;for i6 in L do for j6 in L do if IsZero(i6*j6) then n6:=n6+1;fi;od;od;bA:=0;bO:=0;if IsAbelian(R) then bA:=1;fi;if n3=1 then bO:=1;fi;Print("环R",m,"_",ni,"的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=",Characteristic(R),",",bA,",",bO,",",n1,",",n2,",",n4,",",n5,",",n6,",");n7:=0;;for i7 in L do for j7 in L do if IsZero(i7)=false and IsZero(j7)=false and IsZero(i7*j7) then n7:=n7+1;break;fi;od;od;CR:=Center(R);;n8:=Size(CR);Print(n7,",",n8,"\n");od;
环R12_1的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=12,1,0,12,1,11,11,144,11,12
环R12_2的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=12,1,0,12,2,1,3,60,11,12
环R12_3的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=12,1,0,12,2,5,5,72,11,12
环R12_4的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=12,1,0,12,2,3,3,80,11,12
环R12_5的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=12,1,1,8,4,1,1,40,7,12
环R12_6的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=12,1,0,12,1,5,11,108,11,12
环R12_7的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,1,0,12,1,11,11,144,11,12
环R12_8的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,1,0,12,1,5,11,108,11,12
环R12_9的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,1,0,12,2,5,5,108,11,12
环R12_10的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,0,0,12,3,5,5,90,11,3
环R12_11的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,0,0,12,3,5,5,90,11,3
环R12_12的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,1,0,12,2,5,5,72,11,12
环R12_13的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,1,0,12,4,1,1,60,11,12
环R12_14的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,1,1,10,8,0,0,45,9,12
环R12_15的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,0,0,12,6,1,1,50,11,3
环R12_16的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,1,0,12,2,3,3,80,11,12
环R12_17的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,1,0,12,2,1,3,60,11,12
环R12_18的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,0,0,12,6,1,1,50,7,3
环R12_19的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,1,1,8,4,1,1,40,7,12
环R12_20的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,1,0,12,4,2,2,81,11,12
环R12_21的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,1,0,12,2,2,2,63,11,12
环R12_22的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,1,1,6,4,0,0,35,5,12

GitHub开源项目MathTool地址：https://github.com/Ivanhan2018/MathTool
struct Idx
{
 int n;
 int Id;
 char calR[100];//环的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8
 char name[100];    
};
{4,1,"4,1,0,4,1,3,3,16,3,4","M_4=ring 4.Nu.1"}, \
 {4,2,"4,1,0,4,1,1,3,12,3,4","2Z/8Z=ring 4.Nu.2=Z/8Z的4阶子环{0,2,4,6},运算为模8加与模8乘，是一个幂零理想环"}, \
 {4,3,"4,1,1,2,2,1,1,8,1,4","Z/4Z=ring 4.u.1，运算为模4加与模4乘，有n7=1个零因子，是局部环，极大理想是M_2，(Z/4Z)/M_2=F_2"}, \
 {4,4,"2,1,0,4,1,3,3,16,3,4","M_2×M_2=ring 22.Nu.1"}, \
 {4,5,"2,1,0,4,1,1,3,12,3,4","ring 22.Nu.2=ZimodnZObj(1+i,2+2i)"}, \
 {4,6,"2,1,0,4,2,1,1,12,3,4","M_2×F_2=ring 22.Nu.3"}, \
 {4,7,"2,0,0,4,3,1,1,10,1,1","ring 22.NC.2非交换环最小阶为4，同构意义下只有两个"}, \
 {4,8,"2,0,0,4,3,1,1,10,3,1","ring 22.NC.1"}, \
 {4,9,"2,1,1,2,2,1,1,8,1,4","Z[i]/(2)=F_2[u]/(u^2)=F_2[x,y]/(x^2+1,y+1)=F_2[x,y]/(x^2+1,y)=F_2[x,y,z]/(x+1,y,z^2+1)=ring 22.u.2=F_2+uF_2，有n7=1个零因子，n2=2个幂等元，是介于环Z/4Z与域F_4之间的一种4阶素环、局部环"}, \
 {4,10,"2,1,1,3,4,0,0,9,2,4","4阶布尔环F_2×F_2=F_2[v]/(v(v+1))=F_2[i]=环F2+vF2=ring 22.u.1，有n7=2个零因子，n2=4个幂等元0,1,v,1+v"}, \
 {4,11,"2,1,1,1,2,0,0,7,0,4","F_4=Z[ω]/(2)=F2[x]/(x^2+x+1)=F_2[x,y]/(x^2+x+1,y)=ring 22.u.3，无零因子"}, \
定理：
⑴4阶的交换幺环在同构意义下只有4个：R4_11=F_4、R4_10、R4_9=F2+uF2=ring 22.u.2、R4_3=Z/4Z=ring 4.u.1
⑵4阶局部环只有3个：R4_3、R4_9、R4_11，分别有1个同构于R2_1、R2_1、R2_2=F_2的极大理想，它们都是局部环；R4_10有2个同构于R2_2=F_2的极大理想，它不是局部环。/*
⑶n0=4的4阶循环环有T(4)=|{1,2,4}|=3个:R4_1~R4_3(C、B、A)，n0=2的4阶非循环环有8个:R4_4~R4_11(M2M2、J、Y、M2F2、P、X、D、F2F2、K)
推论：
=>特征为n0=2的4阶有零因子交换幺环有2个：R4_10=环F_2+vF_2、R4_9=F_2+uF_2。
R4_11={0={{0,0},{0,0}},1={{1,0},{0,1}},2={{1,1},{1,0}},3={{0,1},{1,1}}}
R4_10={0=(0,0),3=(1,0),2=(0,1),1=(1,1)}={0={{0,0},{0,0}},1={{1,0},{0,1}},2={{0,0},{0,1}},3={{1,0},{0,0}}}
R4_9={0={{0,0},{0,0}},1={{1,0},{0,1}},2={{0,1},{1,0}},3={{1,1},{1,1}}}
R4_8={0={{0,0},{0,0}},1={{1,0},{0,0}},2={{1,0},{1,0}},3={{0,0},{1,0}}}，是16阶环M_2(Z_2)的一个子环
R4_7={0={{0,0},{0,0}},1={{1,0},{0,0}},2={{1,1},{0,0}},3={{0,1},{0,0}}}，是16阶环M_2(Z_2)的一个子环
R4_2=RingByGenerators([ZmodnZObj(2,8)])=(Z/8Z)的唯一一个极大理想
12阶循环环共有T(12)=|{1,2,3,4,6,12}|=6种，特征都为12。
gap> m:=6;;F:=[ [ ZmodnZObj( 3, m), ZmodnZObj( 1, m) ], [ ZmodnZObj( 0, m ), ZmodnZObj( 3, m ) ] ];;R12_12:=RingByGenerators([F]);;n:=Size(R);;L:=Elements(R);
[ [ [ ZmodnZObj( 0, 6 ), ZmodnZObj( 0, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 0, 6 ) ] ],
  [ [ ZmodnZObj( 0, 6 ), ZmodnZObj( 1, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 0, 6 ) ] ],
  [ [ ZmodnZObj( 0, 6 ), ZmodnZObj( 2, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 0, 6 ) ] ],
  [ [ ZmodnZObj( 0, 6 ), ZmodnZObj( 3, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 0, 6 ) ] ],
  [ [ ZmodnZObj( 0, 6 ), ZmodnZObj( 4, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 0, 6 ) ] ],
  [ [ ZmodnZObj( 0, 6 ), ZmodnZObj( 5, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 0, 6 ) ] ],
  [ [ ZmodnZObj( 3, 6 ), ZmodnZObj( 0, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 3, 6 ) ] ],
  [ [ ZmodnZObj( 3, 6 ), ZmodnZObj( 1, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 3, 6 ) ] ],
  [ [ ZmodnZObj( 3, 6 ), ZmodnZObj( 2, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 3, 6 ) ] ],
  [ [ ZmodnZObj( 3, 6 ), ZmodnZObj( 3, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 3, 6 ) ] ],
  [ [ ZmodnZObj( 3, 6 ), ZmodnZObj( 4, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 3, 6 ) ] ],
  [ [ ZmodnZObj( 3, 6 ), ZmodnZObj( 5, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 3, 6 ) ] ] ]
gap> for i in [1..12] do Print(Size(RingByGenerators([L[i]])),",");od;                                                                        1,6,3,2,3,6,2,12,6,4,6,12,
环R12_12的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,1,0,12,2,5,5,72,11,12

gap> m:=6;;B:=[ [ ZmodnZObj( 1, m), ZmodnZObj( 0, m) ], [ ZmodnZObj( 0, m), ZmodnZObj( 0, m) ] ];;E:=[ [ ZmodnZObj( 3, m), ZmodnZObj( 0, m) ], [ ZmodnZObj( 0, m), ZmodnZObj( 3, m) ] ];;R12_14:=RingByGenerators([B,E]);;n:=Size(R12_14);L:=Elements(R12_14);
12
[ [ [ ZmodnZObj( 0, 6 ), ZmodnZObj( 0, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 0, 6 ) ] ],
  [ [ ZmodnZObj( 0, 6 ), ZmodnZObj( 0, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 3, 6 ) ] ],
  [ [ ZmodnZObj( 1, 6 ), ZmodnZObj( 0, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 0, 6 ) ] ],
  [ [ ZmodnZObj( 1, 6 ), ZmodnZObj( 0, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 3, 6 ) ] ],
  [ [ ZmodnZObj( 2, 6 ), ZmodnZObj( 0, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 0, 6 ) ] ],
  [ [ ZmodnZObj( 2, 6 ), ZmodnZObj( 0, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 3, 6 ) ] ],
  [ [ ZmodnZObj( 3, 6 ), ZmodnZObj( 0, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 0, 6 ) ] ],
  [ [ ZmodnZObj( 3, 6 ), ZmodnZObj( 0, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 3, 6 ) ] ],
  [ [ ZmodnZObj( 4, 6 ), ZmodnZObj( 0, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 0, 6 ) ] ],
  [ [ ZmodnZObj( 4, 6 ), ZmodnZObj( 0, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 3, 6 ) ] ],
  [ [ ZmodnZObj( 5, 6 ), ZmodnZObj( 0, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 0, 6 ) ] ],
  [ [ ZmodnZObj( 5, 6 ), ZmodnZObj( 0, 6 ) ], [ ZmodnZObj( 0, 6 ), ZmodnZObj( 3, 6 ) ] ] ]
gap> m:=12;;ni:=14;;R:=R12_14;;L:=Elements(R);;n1:=0;;for i1 in L do if InverseMutable(i1)=fail then n1:=n1+1;fi;od;n2:=0;;for i2 in L do if IsIdempotent(i2) then n2:=n2+1;fi;od;n3:=0;;for i3 in L do if IsOne(i3) then n3:=n3+1;fi;od;n4:=0;;for i4 in L do if IsZero(i4)=false and i4^2=Zero(R) then n4:=n4+1;fi;od;n5:=0;;for i5 in L do if IsZero(i5)=false and i5^3=Zero(R) then n5:=n5+1;fi;od;n6:=0;;for i6 in L do for j6 in L do if IsZero(i6*j6) then n6:=n6+1;fi;od;od;bA:=0;;bO:=0;;if IsAbelian(R) then bA:=1;fi;if n3=1 then bO:=1;fi;n7:=0;;for i7 in L do for j7 in L do if IsZero(i7)=false and IsZero(j7)=false and IsZero(i7*j7) then n7:=n7+1;break;fi;od;od;CR:=Center(R);;n8:=Size(CR);;Print("环R",m,"_",ni,"的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=",Characteristic(R),",",bA,",",bO,",",n1,",",n2,",",n4,",",n5,",",n6,",",n7,",",n8,"\n");
环R12_14的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,1,0,12,8,0,0,45,9,12【bO,n1=1,10不是0,12】
R12有1个1阶元，3个2阶元，2个3阶元，0个4阶元，6个6阶元，0个12阶元
环的结构不变量n0,bA,bO,n1,n2,n4,n5,n6,n7,n8=6,1,1,10,8,0,0,45,9,12
交换,幺环,不可逆元个数n1=10,幂等元个数n2=8,2次幂零元个数n4=0,2~3次幂零元个数n5=0,零乘个数n6=45,零因子个数n7=9,中心大小n8=12,
gap> L:=Elements(R12_14);;n:=12;;Print("[R",n,"Add]\n");for i1 in L do for i2 in L do Print(Position(L,i1+i2)," ");od;Print("\n");od;Print("[R",n,"Mul]\n");for i1 in L do for i2 in L do Print(Position(L,i1*i2)," ");od;Print("\n");od;                                        [R12Add]
1 2 3 4 5 6 7 8 9 10 11 12
2 1 4 3 6 5 8 7 10 9 12 11
3 4 5 6 7 8 9 10 11 12 1 2
4 3 6 5 8 7 10 9 12 11 2 1
5 6 7 8 9 10 11 12 1 2 3 4
6 5 8 7 10 9 12 11 2 1 4 3
7 8 9 10 11 12 1 2 3 4 5 6
8 7 10 9 12 11 2 1 4 3 6 5
9 10 11 12 1 2 3 4 5 6 7 8
10 9 12 11 2 1 4 3 6 5 8 7
11 12 1 2 3 4 5 6 7 8 9 10
12 11 2 1 4 3 6 5 8 7 10 9
[R12Mul]
1 1 1 1 1 1 1 1 1 1 1 1
1 2 1 2 1 2 1 2 1 2 1 2
1 1 3 3 5 5 7 7 9 9 11 11
1 2 3 4 5 6 7 8 9 10 11 12
1 1 5 5 9 9 1 1 5 5 9 9
1 2 5 6 9 10 1 2 5 6 9 10
1 1 7 7 1 1 7 7 1 1 7 7
1 2 7 8 1 2 7 8 1 2 7 8
1 1 9 9 5 5 1 1 9 9 5 5
1 2 9 10 5 6 1 2 9 10 5 6
1 1 11 11 9 9 7 7 5 5 3 3
1 2 11 12 9 10 7 8 5 6 3 4

20151208：
12阶环分类定理：
R12_1:=DirectSum(SmallRing(3,1),SmallRing(4,1));
R12_6:=DirectSum(SmallRing(3,1),SmallRing(4,2));
R12_3:=DirectSum(SmallRing(3,1),SmallRing(4,3));
R12_7:=DirectSum(SmallRing(3,1),SmallRing(4,4));
R12_7:=DirectSum(SmallRing(2,1),SmallRing(6,1));
R12_8:=DirectSum(SmallRing(3,1),SmallRing(4,5));
R12_9:=DirectSum(SmallRing(3,1),SmallRing(4,6));
R12_9:=DirectSum(SmallRing(2,1),SmallRing(6,3));
R12_9:=DirectSum(SmallRing(2,2),SmallRing(6,1));
R12_10:=DirectSum(SmallRing(3,1),SmallRing(4,7));
R12_11:=DirectSum(SmallRing(3,1),SmallRing(4,8));
R12_12:=DirectSum(SmallRing(3,1),SmallRing(4,9));
R12_20:=DirectSum(SmallRing(3,1),SmallRing(4,10));
R12_20:=DirectSum(SmallRing(2,2),SmallRing(6,3));
R12_21:=DirectSum(SmallRing(3,1),SmallRing(4,11));
R12_4:=DirectSum(SmallRing(3,2),SmallRing(4,1));
R12_2:=DirectSum(SmallRing(3,2),SmallRing(4,2));
R12_5:=DirectSum(SmallRing(3,2),SmallRing(4,3));
R12_16:=DirectSum(SmallRing(3,2),SmallRing(4,4));
R12_16:=DirectSum(SmallRing(2,1),SmallRing(6,2));
R12_17:=DirectSum(SmallRing(3,2),SmallRing(4,5));
R12_13:=DirectSum(SmallRing(3,2),SmallRing(4,6));
R12_13:=DirectSum(SmallRing(2,1),SmallRing(6,4));
R12_13:=DirectSum(SmallRing(2,2),SmallRing(6,2));
R12_18:=DirectSum(SmallRing(3,2),SmallRing(4,7));
R12_15:=DirectSum(SmallRing(3,2),SmallRing(4,8));
R12_19:=DirectSum(SmallRing(3,2),SmallRing(4,9));
R12_14:=DirectSum(SmallRing(3,2),SmallRing(4,10));
R12_14:=DirectSum(SmallRing(2,2),SmallRing(6,4));
R12_22:=DirectSum(SmallRing(3,2),SmallRing(4,11));
gap> for i in [10,11] do R:=SmallRing(12,i);;M:=Ideals(R);;CR:=Center(R);;L:=Elements(R);n8:=0;;for i8 in L do if IsZero(i8)=false and i8^3=Zero(R) then n8:=n8+1;fi;od;Print("i=",i,List(M,Size),Size(CR),",2~4次幂零元个数n8=",n8,"\n");od;
i=10[ 1, 2, 3, 4, 6, 12 ]3,2~4次幂零元个数n8=5
i=11[ 1, 2, 3, 4, 6, 12 ]3,2~4次幂零元个数n8=5
R12_10的中心是3阶零乘环，6阶极大理想是6阶零乘环，4阶理想是R4_7
R12_11的中心是3阶零乘环，6阶极大理想是6阶零乘环，4阶理想是R4_8
6阶零乘环：
不可逆元个数n1=6,幂等元个数n2=1,特征：6,是否交换：true,是否有幺元=false,2次幂零元个数n4=5,2~3次幂零元个数n5=5,零乘个数n6=36,零因子个数n7=5
R4_7：
不可逆元个数n1=4,幂等元个数n2=3,特征：2,是否交换：false,是否有幺元=false,2次幂零元个数n4=1,2~3次幂零元个数n5=1,零乘个数n6=10,零因子个数n7=1
R4_8：
不可逆元个数n1=4,幂等元个数n2=3,特征：2,是否交换：false,是否有幺元=false,2次幂零元个数n4=1,2~3次幂零元个数n5=1,零乘个数n6=10,零因子个数n7=3