无限环
gap> A:=[[1,-1],[-1,1]];;B:=[[1,1],[1,1]];;R:=RingByGenerators([A,B]);;Size(R);IsRing(R);A*B;
infinity
true
[ [ 0, 0 ], [ 0, 0 ] ]
gap> A:=[[0,1],[0,0]];;B:=[[1,0],[0,0]];;C:=[[0,0],[0,1]];;R:=RingByGenerators([A,B,C]);;Size(R);IsRing(R);A*B;B*C;
infinity
true
[ [ 0, 0 ], [ 0, 0 ] ]
[ [ 0, 0 ], [ 0, 0 ] ]
gap> R1:=PolynomialRing(Rationals,1);;x:=Indeterminate(Rationals);;poly:=x^2-3;;IsIrreducibleRingElement(R1,poly);Degree(poly);;Factors(poly);;I:=IdealByGenerators(R1,[poly]);;R:=R1/I;;Size(R);Characteristic(R);IsAbelian(R);
true
infinity
0
true
104种24阶环
gap> NumberSmallRings(18);
0
gap> NumberSmallRings(24);
0
390种16阶环
gap> NumberSmallRings(16);
0//应该是390
16阶循环环共有T(16)=|{1,2,4,8,16}|=5种,特征都为16。
R16_1=Z/256Z_<16>=M_16
R16_2=Z/32Z_<2>
R16_3=Z/48Z_<3>=Z/16Z=Z/16Z_<3>
R16_4=Z/64Z_<4>
R16_5=Z/128Z_<8>
R16_1：2有零因子交换无幺环,1,0,16,1,15,15,256,15,16
R16_2：2有零因子交换无幺环,1,0,16,1,3,7,80,15,16
R16_3：1有零因子交换幺环,1,1,8,2,3,3,48,7,16
R16_4：2有零因子交换无幺环,1,0,16,1,7,15,128,15,16
R16_5：2有零因子交换无幺环,1,0,16,1,7,15,192,15,16
ring 16.u.1=Z_16的运算为模16加与模16乘，剩余类环Z_16是有零因子环。
ring 2222.u.1=F_2×F_2×F_2×F_2，这个环可以看作域F_2上维数是4的向量空间
ring 2222.u.2=F_2×F_2×F_4，这个环是否同构于F_2[u]/(u^4+u)？
ring 2222.u.3=F_2×F_8，这个环是否同构于F_2[u]/(u^4+u^3+u)？
ring 2222.u.4=F_4×F_4，这个环是否同构于F_4[u]/(u^2+u)？
ring 2222.u.5=F_16
R16_100=Z/4Z×Z/4Z：1有零因子交换幺环,1,1,12,4,3,3,64,11,16
256阶全矩阵环M_2(Z/4Z)的一个16阶子环R16_101：4有零因子非交换无幺环,0,0,16,5,5,7,88,7,1
256阶全矩阵环M_2(Z/4Z)的一个16阶子环R16_102：4有零因子非交换无幺环,0,0,16,5,5,7,88,15,1
R16_103：2有零因子交换无幺环,1,0,16,1,3,15,144,15,16
256阶全矩阵环M_2(F_4)的一个16阶子环R16_381：4有零因子非交换无幺环,0,0,16,5,3,3,76,3,1
R16_382=F_2[u]/(u^4+u^3+u^2+u)：1有零因子交换幺环,1,1,12,4,1,3,60,11,16
R16_383=F_2[u]/(u^4+u^2+1)=F_4[u]/(u^2)：1有零因子交换幺环,1,1,4,2,3,3,40,3,16
R16_384=F_2[u]/(u^4+u^2)：1有零因子交换幺环,1,1,12,4,3,3,64,11,16
R16_385=F_2[u]/(u^4)：1有零因子交换幺环,1,1,8,2,3,3,48,7,16
R16_386：1有零因子交换幺环,布尔环,1,1,15,16,0,0,81,14,16
R16_387=F_2[u]/(u^4+u)：1有零因子交换幺环,1,1,13,8,0,0,63,12,16
R16_388=F_2[u]/(u^4+u^3+u)：1有零因子交换幺环,1,1,9,4,0,0,45,8,16
R16_389=F_4[u]/(u^2+u)：1有零因子交换幺环,1,1,7,4,0,0,49,6,16
R16_390=F_2[u]/(u^4+u+1)：0域,1,1,1,2,0,0,31,0,16
16阶全矩阵环M_2(F_2)=R16_300：3有零因子非交换幺环,0,1,10,8,3,3,58,9,2
16阶全矩阵环M_2(M_2)=R16_301：2有零因子交换无幺环,1,0,16,1,15,15,256,15,16 
利用8阶环和2阶环的直积，可以构造出34+24=58种16阶交换非幺环,10种16阶交换幺环,1种16阶非交换幺环：
gap> for i in [1..52] do R:=DirectSum(SmallRing(8,i),SmallRing(2,1));;if IsAbelian(R) and One(R)=fail  then M:=Ideals(R);;Print("i=",i,List(M,Size),"\n");fi;od;
gap> for i in [1..52] do R:=DirectSum(SmallRing(8,i),SmallRing(2,2));;if IsAbelian(R) and One(R)=fail  then M:=Ideals(R);;Print("i=",i,List(M,Size),"\n");fi;od;
gap> for i in [1..52] do R:=DirectSum(SmallRing(8,i),SmallRing(2,1));;if IsAbelian(R) and One(R)<>fail then M:=Ideals(R);;Print("i=",i,List(M,Size),"\n");fi;od;
gap> for i in [1..52] do R:=DirectSum(SmallRing(8,i),SmallRing(2,2));;if IsAbelian(R) and One(R)<>fail then M:=Ideals(R);;Print("i=",i,List(M,Size),"\n");fi;od;
gap> for i in [1..52] do R:=DirectSum(SmallRing(8,i),SmallRing(2,1));;if One(R)<>fail then M:=Ideals(R);;Print("i=",i,List(M,Size),"\n");fi;od;
gap> for i in [1..52] do R:=DirectSum(SmallRing(8,i),SmallRing(2,2));;if One(R)<>fail then M:=Ideals(R);;Print("i=",i,List(M,Size),"\n");fi;od;
问题：16阶全矩阵环M_2(F_2)的4阶子环R4_7、R4_8、R4_9是否是理想子环？
问题：求256阶全矩阵环M_2(Z/4Z)的加法群的所有16阶交换子群。
gap> R1:=PolynomialRing(GF(4),1);;x:=Indeterminate(GF(4));;I1:=IdealByGenerators(R1,[x^2]);;R16_383:=R1/I1;;I2:=IdealByGenerators(R1,[x^2+x]);;R16_389:=R1/I2;;R16_1:=RingByGenerators([ZmodnZObj(16,256)]);;R16_2:=RingByGenerators([ZmodnZObj(2,32)]);;R16_3:=RingByGenerators([ZmodnZObj(3,48)]);;R16_4:=RingByGenerators([ZmodnZObj(4,64)]);;R16_5:=RingByGenerators([ZmodnZObj(8,128)]);;R16_6:=DirectSum(ZmodnZ(8),ZmodnZ(2));;R16_100:=DirectSum(ZmodnZ(4),ZmodnZ(4));;R16_200:=DirectSum(DirectSum(GF(2),GF(2)),ZmodnZ(4));;R16_300:=FullMatrixAlgebra(GF(2),2);;R16:=[R16_1,R16_2,R16_3,R16_4,R16_5,R16_6,R16_100,R16_200,R16_300,R16_383,R16_389];;I:=0;;for R in R16 do I:=I+1;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;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;i:=I;if I=0 then i:=4;fi;Print("I=",i,",特征：",Characteristic(R),",不可逆元个数n1=",n1,",幂等元个 数n2=",n2,",是否交换：",IsAbelian(R),",是否有幺元=",n3=1,",2次幂零元个数n4=",n4,",2~3次幂零元个数n5=",n5,",零乘个数n6=",n6,",零因子个数n7=",n7,"\n");od;
I=1,特征：256,不可逆元个数n1=16,幂等元个数n2=
1,是否交换：true,是否有幺元=false,2次幂零元个数n4=15,2~3次幂零元个数n5=15,零乘个数n6=
256,零因子个数n7=15
I=2,特征：32,不可逆元个数n1=16,幂等元个数n2=
1,是否交换：true,是否有幺元=false,2次幂零元个数n4=3,2~3次幂零元个数n5=7,零乘个数n6=
80,零因子个数n7=15
I=3,特征：48,不可逆元个数n1=16,幂等元个数n2=
2,是否交换：true,是否有幺元=false,2次幂零元个数n4=3,2~3次幂零元个数n5=3,零乘个数n6=
48,零因子个数n7=7
I=4,特征：64,不可逆元个数n1=16,幂等元个数n2=
1,是否交换：true,是否有幺元=false,2次幂零元个数n4=7,2~3次幂零元个数n5=15,零乘个数n6=
128,零因子个数n7=15
I=5,特征：128,不可逆元个数n1=16,幂等元个数n2=
1,是否交换：true,是否有幺元=false,2次幂零元个数n4=7,2~3次幂零元个数n5=15,零乘个数n6=
192,零因子个数n7=15
I=6,特征：fail,不可逆元个数n1=0,幂等元个数n2=
4,是否交换：true,是否有幺元=true,2次幂零元个数n4=1,2~3次幂零元个数n5=3,零乘个数n6=
60,零因子个数n7=11
I=7,特征：fail,不可逆元个数n1=0,幂等元个数n2=
4,是否交换：true,是否有幺元=true,2次幂零元个数n4=3,2~3次幂零元个数n5=3,零乘个数n6=
64,零因子个数n7=11
I=8,特征：fail,不可逆元个数n1=0,幂等元个数n2=
8,是否交换：true,是否有幺元=true,2次幂零元个数n4=1,2~3次幂零元个数n5=1,零乘个数n6=
72,零因子个数n7=13
I=9,特征：2,不可逆元个数n1=10,幂等元个数n2=
8,是否交换：false,是否有幺元=true,2次幂零元个数n4=3,2~3次幂零元个数n5=3,零乘个数n6=
58,零因子个数n7=9
I=10,特征：2,不可逆元个数n1=4,幂等元个数n2=
2,是否交换：true,是否有幺元=true,2次幂零元个数n4=3,2~3次幂零元个数n5=3,零乘个数n6=
40,零因子个数n7=3
I=11,特征：2,不可逆元个数n1=7,幂等元个数n2=
4,是否交换：true,是否有幺元=true,2次幂零元个数n4=0,2~3次幂零元个数n5=0,零乘个数n6=
49,零因子个数n7=6
gap> R1:=PolynomialRing(GF(4),1);;x:=Indeterminate(GF(4));;I1:=IdealByGenerators(R1,[x^2]);;R16_383:=R1/I1;;I2:=IdealByGenerators(R1,[x^2+x]);;R16_389:=R1/I2;;R16_1:=RingByGenerators([ZmodnZObj(16,256)]);;R16_2:=RingByGenerators([ZmodnZObj(2,32)]);;R16_3:=RingByGenerators([ZmodnZObj(3,48)]);;R16_4:=RingByGenerators([ZmodnZObj(4,64)]);;R16_5:=RingByGenerators([ZmodnZObj(8,128)]);;R16_6:=DirectSum(ZmodnZ(8),ZmodnZ(2));;R16_100:=DirectSum(ZmodnZ(4),ZmodnZ(4));;R16_200:=DirectSum(DirectSum(GF(2),GF(2)),ZmodnZ(4));;R16_300:=FullMatrixAlgebra(GF(2),2);;R16:=[R16_1,R16_2,R16_3,R16_4,R16_5,R16_6,R16_100,R16_200,R16_300,R16_383,R16_389];;I:=0;;for R in R16 do I:=I+1;L:=Elements(R);n:=Size(R);L:=Elements(R);;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;od;
[R16Add]
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,
3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,2,
4,5,6,7,8,9,10,11,12,13,14,15,16,1,2,3,
5,6,7,8,9,10,11,12,13,14,15,16,1,2,3,4,
6,7,8,9,10,11,12,13,14,15,16,1,2,3,4,5,
7,8,9,10,11,12,13,14,15,16,1,2,3,4,5,6,
8,9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,
9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,
10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,9,
11,12,13,14,15,16,1,2,3,4,5,6,7,8,9,10,
12,13,14,15,16,1,2,3,4,5,6,7,8,9,10,11,
13,14,15,16,1,2,3,4,5,6,7,8,9,10,11,12,
14,15,16,1,2,3,4,5,6,7,8,9,10,11,12,13,
15,16,1,2,3,4,5,6,7,8,9,10,11,12,13,14,
16,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,
[R16Mul]
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
[R16Add]
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,
3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,2,
4,5,6,7,8,9,10,11,12,13,14,15,16,1,2,3,
5,6,7,8,9,10,11,12,13,14,15,16,1,2,3,4,
6,7,8,9,10,11,12,13,14,15,16,1,2,3,4,5,
7,8,9,10,11,12,13,14,15,16,1,2,3,4,5,6,
8,9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,
9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,
10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,9,
11,12,13,14,15,16,1,2,3,4,5,6,7,8,9,10,
12,13,14,15,16,1,2,3,4,5,6,7,8,9,10,11,
13,14,15,16,1,2,3,4,5,6,7,8,9,10,11,12,
14,15,16,1,2,3,4,5,6,7,8,9,10,11,12,13,
15,16,1,2,3,4,5,6,7,8,9,10,11,12,13,14,
16,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,
[R16Mul]
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,3,5,7,9,11,13,15,1,3,5,7,9,11,13,15,
1,5,9,13,1,5,9,13,1,5,9,13,1,5,9,13,
1,7,13,3,9,15,5,11,1,7,13,3,9,15,5,11,
1,9,1,9,1,9,1,9,1,9,1,9,1,9,1,9,
1,11,5,15,9,3,13,7,1,11,5,15,9,3,13,7,
1,13,9,5,1,13,9,5,1,13,9,5,1,13,9,5,
1,15,13,11,9,7,5,3,1,15,13,11,9,7,5,3,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,3,5,7,9,11,13,15,1,3,5,7,9,11,13,15,
1,5,9,13,1,5,9,13,1,5,9,13,1,5,9,13,
1,7,13,3,9,15,5,11,1,7,13,3,9,15,5,11,
1,9,1,9,1,9,1,9,1,9,1,9,1,9,1,9,
1,11,5,15,9,3,13,7,1,11,5,15,9,3,13,7,
1,13,9,5,1,13,9,5,1,13,9,5,1,13,9,5,
1,15,13,11,9,7,5,3,1,15,13,11,9,7,5,3,
[R16Add]
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,
3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,2,
4,5,6,7,8,9,10,11,12,13,14,15,16,1,2,3,
5,6,7,8,9,10,11,12,13,14,15,16,1,2,3,4,
6,7,8,9,10,11,12,13,14,15,16,1,2,3,4,5,
7,8,9,10,11,12,13,14,15,16,1,2,3,4,5,6,
8,9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,
9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,
10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,9,
11,12,13,14,15,16,1,2,3,4,5,6,7,8,9,10,
12,13,14,15,16,1,2,3,4,5,6,7,8,9,10,11,
13,14,15,16,1,2,3,4,5,6,7,8,9,10,11,12,
14,15,16,1,2,3,4,5,6,7,8,9,10,11,12,13,
15,16,1,2,3,4,5,6,7,8,9,10,11,12,13,14,
16,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,
[R16Mul]
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,4,7,10,13,16,3,6,9,12,15,2,5,8,11,14,
1,7,13,3,9,15,5,11,1,7,13,3,9,15,5,11,
1,10,3,12,5,14,7,16,9,2,11,4,13,6,15,8,
1,13,9,5,1,13,9,5,1,13,9,5,1,13,9,5,
1,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,
1,3,5,7,9,11,13,15,1,3,5,7,9,11,13,15,
1,6,11,16,5,10,15,4,9,14,3,8,13,2,7,12,
1,9,1,9,1,9,1,9,1,9,1,9,1,9,1,9,
1,12,7,2,13,8,3,14,9,4,15,10,5,16,11,6,
1,15,13,11,9,7,5,3,1,15,13,11,9,7,5,3,
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
1,5,9,13,1,5,9,13,1,5,9,13,1,5,9,13,
1,8,15,6,13,4,11,2,9,16,7,14,5,12,3,10,
1,11,5,15,9,3,13,7,1,11,5,15,9,3,13,7,
1,14,11,8,5,2,15,12,9,6,3,16,13,10,7,4,
[R16Add]
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,
3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,2,
4,5,6,7,8,9,10,11,12,13,14,15,16,1,2,3,
5,6,7,8,9,10,11,12,13,14,15,16,1,2,3,4,
6,7,8,9,10,11,12,13,14,15,16,1,2,3,4,5,
7,8,9,10,11,12,13,14,15,16,1,2,3,4,5,6,
8,9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,
9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,
10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,9,
11,12,13,14,15,16,1,2,3,4,5,6,7,8,9,10,
12,13,14,15,16,1,2,3,4,5,6,7,8,9,10,11,
13,14,15,16,1,2,3,4,5,6,7,8,9,10,11,12,
14,15,16,1,2,3,4,5,6,7,8,9,10,11,12,13,
15,16,1,2,3,4,5,6,7,8,9,10,11,12,13,14,
16,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,
[R16Mul]
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,5,9,13,1,5,9,13,1,5,9,13,1,5,9,13,
1,9,1,9,1,9,1,9,1,9,1,9,1,9,1,9,
1,13,9,5,1,13,9,5,1,13,9,5,1,13,9,5,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,5,9,13,1,5,9,13,1,5,9,13,1,5,9,13,
1,9,1,9,1,9,1,9,1,9,1,9,1,9,1,9,
1,13,9,5,1,13,9,5,1,13,9,5,1,13,9,5,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,5,9,13,1,5,9,13,1,5,9,13,1,5,9,13,
1,9,1,9,1,9,1,9,1,9,1,9,1,9,1,9,
1,13,9,5,1,13,9,5,1,13,9,5,1,13,9,5,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,5,9,13,1,5,9,13,1,5,9,13,1,5,9,13,
1,9,1,9,1,9,1,9,1,9,1,9,1,9,1,9,
1,13,9,5,1,13,9,5,1,13,9,5,1,13,9,5,
[R16Add]
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,
3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,2,
4,5,6,7,8,9,10,11,12,13,14,15,16,1,2,3,
5,6,7,8,9,10,11,12,13,14,15,16,1,2,3,4,
6,7,8,9,10,11,12,13,14,15,16,1,2,3,4,5,
7,8,9,10,11,12,13,14,15,16,1,2,3,4,5,6,
8,9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,
9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,
10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,9,
11,12,13,14,15,16,1,2,3,4,5,6,7,8,9,10,
12,13,14,15,16,1,2,3,4,5,6,7,8,9,10,11,
13,14,15,16,1,2,3,4,5,6,7,8,9,10,11,12,
14,15,16,1,2,3,4,5,6,7,8,9,10,11,12,13,
15,16,1,2,3,4,5,6,7,8,9,10,11,12,13,14,
16,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,
[R16Mul]
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,9,1,9,1,9,1,9,1,9,1,9,1,9,1,9,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,9,1,9,1,9,1,9,1,9,1,9,1,9,1,9,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,9,1,9,1,9,1,9,1,9,1,9,1,9,1,9,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,9,1,9,1,9,1,9,1,9,1,9,1,9,1,9,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,9,1,9,1,9,1,9,1,9,1,9,1,9,1,9,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,9,1,9,1,9,1,9,1,9,1,9,1,9,1,9,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,9,1,9,1,9,1,9,1,9,1,9,1,9,1,9,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,9,1,9,1,9,1,9,1,9,1,9,1,9,1,9,
[R16Add]
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,
3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,2,
4,3,6,5,8,7,10,9,12,11,14,13,16,15,2,1,
5,6,7,8,9,10,11,12,13,14,15,16,1,2,3,4,
6,5,8,7,10,9,12,11,14,13,16,15,2,1,4,3,
7,8,9,10,11,12,13,14,15,16,1,2,3,4,5,6,
8,7,10,9,12,11,14,13,16,15,2,1,4,3,6,5,
9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,
10,9,12,11,14,13,16,15,2,1,4,3,6,5,8,7,
11,12,13,14,15,16,1,2,3,4,5,6,7,8,9,10,
12,11,14,13,16,15,2,1,4,3,6,5,8,7,10,9,
13,14,15,16,1,2,3,4,5,6,7,8,9,10,11,12,
14,13,16,15,2,1,4,3,6,5,8,7,10,9,12,11,
15,16,1,2,3,4,5,6,7,8,9,10,11,12,13,14,
16,15,2,1,4,3,6,5,8,7,10,9,12,11,14,13,
[R16Mul]
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,
1,1,3,3,5,5,7,7,9,9,11,11,13,13,15,15,
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
1,1,5,5,9,9,13,13,1,1,5,5,9,9,13,13,
1,2,5,6,9,10,13,14,1,2,5,6,9,10,13,14,
1,1,7,7,13,13,3,3,9,9,15,15,5,5,11,11,
1,2,7,8,13,14,3,4,9,10,15,16,5,6,11,12,
1,1,9,9,1,1,9,9,1,1,9,9,1,1,9,9,
1,2,9,10,1,2,9,10,1,2,9,10,1,2,9,10,
1,1,11,11,5,5,15,15,9,9,3,3,13,13,7,7,
1,2,11,12,5,6,15,16,9,10,3,4,13,14,7,8,
1,1,13,13,9,9,5,5,1,1,13,13,9,9,5,5,
1,2,13,14,9,10,5,6,1,2,13,14,9,10,5,6,
1,1,15,15,13,13,11,11,9,9,7,7,5,5,3,3,
1,2,15,16,13,14,11,12,9,10,7,8,5,6,3,4,
[R16Add]
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
2,3,4,1,6,7,8,5,10,11,12,9,14,15,16,13,
3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14,
4,1,2,3,8,5,6,7,12,9,10,11,16,13,14,15,
5,6,7,8,9,10,11,12,13,14,15,16,1,2,3,4,
6,7,8,5,10,11,12,9,14,15,16,13,2,3,4,1,
7,8,5,6,11,12,9,10,15,16,13,14,3,4,1,2,
8,5,6,7,12,9,10,11,16,13,14,15,4,1,2,3,
9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,
10,11,12,9,14,15,16,13,2,3,4,1,6,7,8,5,
11,12,9,10,15,16,13,14,3,4,1,2,7,8,5,6,
12,9,10,11,16,13,14,15,4,1,2,3,8,5,6,7,
13,14,15,16,1,2,3,4,5,6,7,8,9,10,11,12,
14,15,16,13,2,3,4,1,6,7,8,5,10,11,12,9,
15,16,13,14,3,4,1,2,7,8,5,6,11,12,9,10,
16,13,14,15,4,1,2,3,8,5,6,7,12,9,10,11,
[R16Mul]
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,2,3,4,1,2,3,4,1,2,3,4,1,2,3,4,
1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,
1,4,3,2,1,4,3,2,1,4,3,2,1,4,3,2,
1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
1,3,1,3,5,7,5,7,9,11,9,11,13,15,13,15,
1,4,3,2,5,8,7,6,9,12,11,10,13,16,15,14,
1,1,1,1,9,9,9,9,1,1,1,1,9,9,9,9,
1,2,3,4,9,10,11,12,1,2,3,4,9,10,11,12,
1,3,1,3,9,11,9,11,1,3,1,3,9,11,9,11,
1,4,3,2,9,12,11,10,1,4,3,2,9,12,11,10,
1,1,1,1,13,13,13,13,9,9,9,9,5,5,5,5,
1,2,3,4,13,14,15,16,9,10,11,12,5,6,7,8,
1,3,1,3,13,15,13,15,9,11,9,11,5,7,5,7,
1,4,3,2,13,16,15,14,9,12,11,10,5,8,7,6,
[R16Add]
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
2,3,4,1,6,7,8,5,10,11,12,9,14,15,16,13,
3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14,
4,1,2,3,8,5,6,7,12,9,10,11,16,13,14,15,
5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12,
6,7,8,5,2,3,4,1,14,15,16,13,10,11,12,9,
7,8,5,6,3,4,1,2,15,16,13,14,11,12,9,10,
8,5,6,7,4,1,2,3,16,13,14,15,12,9,10,11,
9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,
10,11,12,9,14,15,16,13,2,3,4,1,6,7,8,5,
11,12,9,10,15,16,13,14,3,4,1,2,7,8,5,6,
12,9,10,11,16,13,14,15,4,1,2,3,8,5,6,7,
13,14,15,16,9,10,11,12,5,6,7,8,1,2,3,4,
14,15,16,13,10,11,12,9,6,7,8,5,2,3,4,1,
15,16,13,14,11,12,9,10,7,8,5,6,3,4,1,2,
16,13,14,15,12,9,10,11,8,5,6,7,4,1,2,3,
[R16Mul]
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,2,3,4,1,2,3,4,1,2,3,4,1,2,3,4,
1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,
1,4,3,2,1,4,3,2,1,4,3,2,1,4,3,2,
1,1,1,1,5,5,5,5,1,1,1,1,5,5,5,5,
1,2,3,4,5,6,7,8,1,2,3,4,5,6,7,8,
1,3,1,3,5,7,5,7,1,3,1,3,5,7,5,7,
1,4,3,2,5,8,7,6,1,4,3,2,5,8,7,6,
1,1,1,1,1,1,1,1,9,9,9,9,9,9,9,9,
1,2,3,4,1,2,3,4,9,10,11,12,9,10,11,12,
1,3,1,3,1,3,1,3,9,11,9,11,9,11,9,11,
1,4,3,2,1,4,3,2,9,12,11,10,9,12,11,10,
1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
1,3,1,3,5,7,5,7,9,11,9,11,13,15,13,15,
1,4,3,2,5,8,7,6,9,12,11,10,13,16,15,14,
[R16Add]
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,
3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14,
4,3,2,1,8,7,6,5,12,11,10,9,16,15,14,13,
5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12,
6,5,8,7,2,1,4,3,14,13,16,15,10,9,12,11,
7,8,5,6,3,4,1,2,15,16,13,14,11,12,9,10,
8,7,6,5,4,3,2,1,16,15,14,13,12,11,10,9,
9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,
10,9,12,11,14,13,16,15,2,1,4,3,6,5,8,7,
11,12,9,10,15,16,13,14,3,4,1,2,7,8,5,6,
12,11,10,9,16,15,14,13,4,3,2,1,8,7,6,5,
13,14,15,16,9,10,11,12,5,6,7,8,1,2,3,4,
14,13,16,15,10,9,12,11,6,5,8,7,2,1,4,3,
15,16,13,14,11,12,9,10,7,8,5,6,3,4,1,2,
16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,
[R16Mul]
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,2,3,4,1,2,3,4,1,2,3,4,1,2,3,4,
1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,
1,2,3,4,2,1,4,3,3,4,1,2,4,3,2,1,
1,5,9,13,1,5,9,13,1,5,9,13,1,5,9,13,
1,6,11,16,1,6,11,16,1,6,11,16,1,6,11,16,
1,5,9,13,2,6,10,14,3,7,11,15,4,8,12,16,
1,6,11,16,2,5,12,15,3,8,9,14,4,7,10,13,
1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
1,1,1,1,6,6,6,6,11,11,11,11,16,16,16,16,
1,2,3,4,6,5,8,7,11,12,9,10,16,15,14,13,
1,5,9,13,5,1,13,9,9,13,1,5,13,9,5,1,
1,6,11,16,5,2,15,12,9,14,3,8,13,10,7,4,
1,5,9,13,6,2,14,10,11,15,3,7,16,12,8,4,
1,6,11,16,6,1,16,11,11,16,1,6,16,11,6,1,
[R16Add]
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,
3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14,
4,3,2,1,8,7,6,5,12,11,10,9,16,15,14,13,
5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12,
6,5,8,7,2,1,4,3,14,13,16,15,10,9,12,11,
7,8,5,6,3,4,1,2,15,16,13,14,11,12,9,10,
8,7,6,5,4,3,2,1,16,15,14,13,12,11,10,9,
9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,
10,9,12,11,14,13,16,15,2,1,4,3,6,5,8,7,
11,12,9,10,15,16,13,14,3,4,1,2,7,8,5,6,
12,11,10,9,16,15,14,13,4,3,2,1,8,7,6,5,
13,14,15,16,9,10,11,12,5,6,7,8,1,2,3,4,
14,13,16,15,10,9,12,11,6,5,8,7,2,1,4,3,
15,16,13,14,11,12,9,10,7,8,5,6,3,4,1,2,
16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,
[R16Mul]
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,
1,1,1,1,3,3,3,3,4,4,4,4,2,2,2,2,
1,1,1,1,4,4,4,4,2,2,2,2,3,3,3,3,
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
1,2,3,4,6,5,8,7,11,12,9,10,16,15,14,13,
1,2,3,4,7,8,5,6,12,11,10,9,14,13,16,15,
1,2,3,4,8,7,6,5,10,9,12,11,15,16,13,14,
1,3,4,2,9,11,12,10,13,15,16,14,5,7,8,6,
1,3,4,2,10,12,11,9,15,13,14,16,8,6,5,7,
1,3,4,2,11,9,10,12,16,14,13,15,6,8,7,5,
1,3,4,2,12,10,9,11,14,16,15,13,7,5,6,8,
1,4,2,3,13,16,14,15,5,8,6,7,9,12,10,11,
1,4,2,3,14,15,13,16,7,6,8,5,12,9,11,10,
1,4,2,3,15,14,16,13,8,5,7,6,10,11,9,12,
1,4,2,3,16,13,15,14,6,7,5,8,11,10,12,9,
[R16Add]
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,
3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14,
4,3,2,1,8,7,6,5,12,11,10,9,16,15,14,13,
5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12,
6,5,8,7,2,1,4,3,14,13,16,15,10,9,12,11,
7,8,5,6,3,4,1,2,15,16,13,14,11,12,9,10,
8,7,6,5,4,3,2,1,16,15,14,13,12,11,10,9,
9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,
10,9,12,11,14,13,16,15,2,1,4,3,6,5,8,7,
11,12,9,10,15,16,13,14,3,4,1,2,7,8,5,6,
12,11,10,9,16,15,14,13,4,3,2,1,8,7,6,5,
13,14,15,16,9,10,11,12,5,6,7,8,1,2,3,4,
14,13,16,15,10,9,12,11,6,5,8,7,2,1,4,3,
15,16,13,14,11,12,9,10,7,8,5,6,3,4,1,2,
16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,
[R16Mul]
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,2,3,4,2,1,4,3,3,4,1,2,4,3,2,1,
1,3,4,2,3,1,2,4,4,2,1,3,2,4,3,1,
1,4,2,3,4,1,3,2,2,3,1,4,3,2,4,1,
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
1,1,1,1,6,6,6,6,11,11,11,11,16,16,16,16,
1,4,2,3,7,6,8,5,12,9,11,10,14,15,13,16,
1,3,4,2,8,6,5,7,10,12,11,9,15,13,14,16,
1,3,4,2,9,11,12,10,13,15,16,14,5,7,8,6,
1,4,2,3,10,11,9,12,15,14,16,13,8,5,7,6,
1,1,1,1,11,11,11,11,16,16,16,16,6,6,6,6,
1,2,3,4,12,11,10,9,14,13,16,15,7,8,5,6,
1,4,2,3,13,16,14,15,5,8,6,7,9,12,10,11,
1,3,4,2,14,16,15,13,7,5,6,8,12,10,9,11,
1,2,3,4,15,16,13,14,8,7,6,5,10,9,12,11,
1,1,1,1,16,16,16,16,6,6,6,6,11,11,11,11,

R16_381
gap> F4:=Elements(GF(4));A:=[[F4[2],F4[1]], [F4[2],F4[1]]];B:=[[F4[2],F4[4]], [F4[2],F4[4]]];R:=RingByGenerators([A,B]);;Size(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;Print("不可逆元个数n1=",n1,",幂等元个数n2=",n2,",特征：",Characteristic(R),",是否交  ：",IsAbelian(R),"\n");n:=Size(R);L:=Elements(R);;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;
[ 0*Z(2), Z(2)^0, Z(2^2), Z(2^2)^2 ]
[ [ Z(2)^0, 0*Z(2) ], [ Z(2)^0, 0*Z(2) ] ]
[ [ Z(2)^0, Z(2^2)^2 ], [ Z(2)^0, Z(2^2)^2 ] ]
16
[ [ [ 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2) ] ], [ [ 0*Z(2), Z(2)^0 ], [ 0*Z(2), Z(2)^0 ] ],
  [ [ 0*Z(2), Z(2^2) ], [ 0*Z(2), Z(2^2) ] ], [ [ 0*Z(2), Z(2^2)^2 ], [ 0*Z(2), Z(2^2)^2 ] ],
  [ [ Z(2)^0, 0*Z(2) ], [ Z(2)^0, 0*Z(2) ] ], [ [ Z(2)^0, Z(2)^0 ], [ Z(2)^0, Z(2)^0 ] ],
  [ [ Z(2)^0, Z(2^2) ], [ Z(2)^0, Z(2^2) ] ], [ [ Z(2)^0, Z(2^2)^2 ], [ Z(2)^0, Z(2^2)^2 ] ],
  [ [ Z(2^2), 0*Z(2) ], [ Z(2^2), 0*Z(2) ] ], [ [ Z(2^2), Z(2)^0 ], [ Z(2^2), Z(2)^0 ] ],
  [ [ Z(2^2), Z(2^2) ], [ Z(2^2), Z(2^2) ] ], [ [ Z(2^2), Z(2^2)^2 ], [ Z(2^2), Z(2^2)^2 ] ],
  [ [ Z(2^2)^2, 0*Z(2) ], [ Z(2^2)^2, 0*Z(2) ] ], [ [ Z(2^2)^2, Z(2)^0 ], [ Z(2^2)^2, Z(2)^0 ] ],
  [ [ Z(2^2)^2, Z(2^2) ], [ Z(2^2)^2, Z(2^2) ] ], [ [ Z(2^2)^2, Z(2^2)^2 ], [ Z(2^2)^2, Z(2^2)^2 ] ] ]
不可逆元个数n1=16,幂等元个数n2=5,特征：2,是否交换：false
16
[R16Add]
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,
3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14,
4,3,2,1,8,7,6,5,12,11,10,9,16,15,14,13,
5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12,
6,5,8,7,2,1,4,3,14,13,16,15,10,9,12,11,
7,8,5,6,3,4,1,2,15,16,13,14,11,12,9,10,
8,7,6,5,4,3,2,1,16,15,14,13,12,11,10,9,
9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8,
10,9,12,11,14,13,16,15,2,1,4,3,6,5,8,7,
11,12,9,10,15,16,13,14,3,4,1,2,7,8,5,6,
12,11,10,9,16,15,14,13,4,3,2,1,8,7,6,5,
13,14,15,16,9,10,11,12,5,6,7,8,1,2,3,4,
14,13,16,15,10,9,12,11,6,5,8,7,2,1,4,3,
15,16,13,14,11,12,9,10,7,8,5,6,3,4,1,2,
16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,
[R16Mul]
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
1,3,4,2,9,11,12,10,13,15,16,14,5,7,8,6,
1,4,2,3,13,16,14,15,5,8,6,7,9,12,10,11,
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,4,2,3,13,16,14,15,5,8,6,7,9,12,10,11,
1,3,4,2,9,11,12,10,13,15,16,14,5,7,8,6,
1,3,4,2,9,11,12,10,13,15,16,14,5,7,8,6,
1,4,2,3,13,16,14,15,5,8,6,7,9,12,10,11,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
1,4,2,3,13,16,14,15,5,8,6,7,9,12,10,11,
1,3,4,2,9,11,12,10,13,15,16,14,5,7,8,6,
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

特征为8、加法群为C_8×C_2的16阶环分配编号空间6~100（近似上限值）
特征为4、加法群为C_4×C_4的16阶环分配编号空间100~200（近似上、下限值）
特征为4、加法群为C_4×C_2×C_2的16阶环分配编号空间200~300（近似上、下限值）
特征为2的16阶环分配编号空间300~390（近似下限值）
12阶环、2阶环的直积共有22*2=44种
8阶环、3阶环的直积共有52*2=104种
4阶环、6阶环的直积共有11*4=44种
美国数学家纳坦·雅各布森（Nathan Jacobson，1910.10.5-1999.12.5，又译作雅可布孙、贾克勃逊）对于半单环的分类，创立了他的结构理论。他认为对任意环均可引进根基的概念，而对阿廷环来说，根基就是一组真幂零元。
有限半单环的个数：
1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 6, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 13, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 6, 6, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 8, 1, 2, 2
No：
1, 1, 2, 3, 6, 8, 13, 18, 29, 40, 58, 79, 115, 154, 213, 284, 391, 514, 690, 900, 1197, 1549, 2025, 2600, 3377, 4306, 5523, 7000, 8922, 11235, 14196, 17777, 22336, 27825, 34720, 43037, 53446, 65942, 81423, 100033, 122991, 150481, 184149, 224449, 273614, 332291
有限非交换环的个数：
0, 0, 0, 2, 0, 0, 0, 18, 2, 0, 0, 4, 0, 0, 0, 228, 0, 4, 0, 4, 0, 0, 0, 36, 2, 0, 23, 4, 0, 0, 0
有限交换环的个数：
1, 2, 2, 9, 2, 4, 2, 34, 9, 4, 2, 18, 2, 4, 4, 162, 2, 18, 2, 18, 4, 4, 2, 68, 9, 4, 36, 18, 2, 8, 2
有限幺环的个数：
1, 1, 1, 4, 1, 1, 1, 11, 4, 1, 1, 4, 1, 1, 1, 50, 1, 4, 1, 4, 1, 1, 1, 11, 4, 1, 12, 4, 1, 1, 1, 208, 1, 1, 1, 16, 1, 1, 1, 11, 1, 1, 1, 4, 4, 1, 1, 50, 4, 4, 1, 4, 1, 11, 1, 11, 1, 1, 1, 4, 1, 1, 4