1. Кострикин А.И. Введение в алгебру. – М.: Наука, 1997.
2. Скорняков Л.А. Элементы алгебры. – М.: Наука, 1986.
Приложение
1. Текст программы поиска колец, имеющих аддитивную группу 
.
Программа 1. Формирование мультипликативных полугрупп:
program form;
uses crt;
type mas=array [0..7,0..7] of integer;
dmas=array [1..7,1..7] of integer;
var a:mas;
b:dmas;
f:text;
procedure sc(var a:mas);
var i,j:integer;
begin
assign(f,'a:\mat.txt');
reset(f);
for i:=0 to 7 do begin
for j:=0 to 7 do read(f,a[i,j]);
readln(f);
end;
close(f);
end;
procedure zap1(k1,k2,k3,k4:integer;a:mas; var b:dmas);
var t2,t3,t5,t6,t7:integer;
begin
b[1,1]:=k1; b[1,4]:=k2; b[4,1]:=k3; b[4,4]:=k4;
b[1,2]:=a[b[1,1],b[1,1]]; b[1,3]:=a[b[1,1],b[1,2]];
b[1,5]:=a[b[1,1],b[1,4]]; b[1,6]:=a[b[1,1],b[1,5]];
b[1,7]:=a[b[1,1],b[1,6]];
for t2:=1 to 7 do b[2,t2]:=a[b[1,t2],b[1,t2]];
for t3:=1 to 7 do b[3,t3]:=a[b[1,t3],b[2,t3]];
b[4,2]:=a[b[4,1],b[4,1]]; b[4,3]:=a[b[4,1],b[4,2]];
b[4,5]:=a[b[4,1],b[4,4]]; b[4,6]:=a[b[4,1],b[4,5]];
b[4,7]:=a[b[4,1],b[4,6]];
for t5:=1 to 7 do b[5,t5]:=a[b[1,t5],b[4,t5]];
for t6:=1 to 7 do b[6,t6]:=a[b[1,t6],b[5,t6]];
for t7:=1 to 7 do b[7,t7]:=a[b[1,t7],b[6,t7]];
end;
procedure zap(a:mas);
var f1:text;
i,j,k1,k2,k3,k4,r:integer;
begin
assign(f1,'a:\a2.txt');
rewrite(f1);
r:=0;
for k1:=0 to 7 do
for k2:=0 to 7 do
for k3:=0 to 7 do
for k4:=0 to 7 do begin
zap1(k1,k2,k3,k4,a,b);
for i:=1 to 7 do begin
for j:=1 to 7 do write(f1,b[i,j]);
writeln(f1);
end;
writeln(f1); inc(r);
end;
close(f1);
writeln(r);
end;
begin
clrscr;
sc(a);
zap(a);
writeln('ok');
readln;
end.
Программа 2. Проверка на ассоциативность:
program assoc;
uses crt;
type mas=array [0..7,0..7] of integer;
var a,b:mas;
i,j,k1,k2,k3,n,i1,i2,r,q:integer;
f,f1:text;
begin
clrscr;
q:=0;
assign(f,'a:\w03.txt');
reset(f);
assign(f1,'a:\aw03.txt');
rewrite(f1);
n:=1; r:=0;
while n<=2048 do begin
for i:=0 to 7 do begin
for j:=0 to 7 do read(f,a[i,j]);
readln(f);
end;
for k1:=1 to 7 do
for k2:=1 to 7 do
for k3:=1 to 7 do
if a[k1,a[k2,k3]]<>a[a[k1,k2],k3] then q:=1;
if q<>1 then begin
for i1:=0 to 7 do begin
for i2:=0 to 7 do write(f1,a[i1,i2],' ');
writeln(f1);
end;
inc(r);
writeln(f1);
end;
readln(f);inc(n);q:=0;
end;
close(f);
close(f1);
writeln(r);
writeln('ok');
readln;
end.
Программа 3. Нахождение изоморфных колец:
program izomorf;
uses crt;
const p=1;
type mas=array [0..7,0..7] of integer;
mas1=array [1..8,1..8] of integer;
var a,b:mas;
y:mas1;
i,j,n,i1,i2,r,w,h,g,d:integer;
f,f1,f2:text;
label 1;
procedure alfa(a:mas; var b:mas);
var k1,k2,j1,j2,z1,z2,z3,z4,q3,w3:integer;
begin
for k1:=0 to 7 do begin
b[0,k1]:=0;
b[1,k1]:=a[7,k1];
b[2,k1]:=a[2,k1];
b[3,k1]:=a[5,k1];
b[4,k1]:=a[4,k1];
b[5,k1]:=a[3,k1];
b[6,k1]:=a[6,k1];
b[7,k1]:=a[1,k1];
end;
for k2:=0 to 7 do begin
z1:=b[k2,3];z2:=b[k2,5];
b[k2,5]:=z1;b[k2,3]:=z2;
z3:=b[k2,1];z4:=b[k2,7];
b[k2,7]:=z3;b[k2,1]:=z4;
end;
for j1:=0 to 7 do
for j2:=0 to 7 do begin
if b[j1,j2]=1 then b[j1,j2]:=-1;
if b[j1,j2]=7 then b[j1,j2]:=-7;
end;
for j1:=0 to 7 do
for j2:=0 to 7 do begin
if b[j1,j2]=-1 then b[j1,j2]:=7;
if b[j1,j2]=-7 then b[j1,j2]:=1;
end;
for j1:=0 to 7 do
for j2:=0 to 7 do begin
if b[j1,j2]=5 then b[j1,j2]:=-5;
if b[j1,j2]=3 then b[j1,j2]:=-3;
end;
for j1:=0 to 7 do
for j2:=0 to 7 do begin
if b[j1,j2]=-5 then b[j1,j2]:=3;
if b[j1,j2]=-3 then b[j1,j2]:=5;
end;
for q3:=0 to 7 do begin
for w3:=0 to 7 do write(b[q3,w3],' ');
writeln;
end;
end;
procedure prov(b:mas);
var q,i3,j3,t,m,x,i4,j4:integer;
a1:mas;
begin
t:=1;
reset(f);
rewrite(f1);
for m:=1 to 9 do readln(f);
while t<=p do begin
for i3:=0 to 7 do begin
for j3:=0 to 7 do read(f,a1[i3,j3]);
readln(f);
end;
q:=0;
for i3:=0 to 7 do
for j3:=0 to 7 do begin
if a1[i3,j3]=b[i3,j3] then inc(q); end;
if q<>64 then
{for x:=1 to 9 do readln(f) else}
begin
for i4:=0 to 7 do begin
for j4:=0 to 7 do write(f1,a1[i4,j4],' ');
writeln(f1);
end;
writeln(f1);
end;
readln(f);
inc(t);
end;
end;
begin
clrscr;
assign(f1,'a:\pr.txt');
assign(f2,'a:\iaw.txt');
rewrite(f2);
assign(f,'a:\aw03.txt');
n:=1;
reset(f);
rewrite(f1);
for i:=0 to 7 do begin
for j:=0 to 7 do read(f,a[i,j]);
readln(f);
end;
{if a[i,j]=8 then begin n:=8; goto 1; end;}
alfa(a,b);
readln(f);
w:=0;
for h:=0 to 7 do
for g:=0 to 7 do
if a[h,g]=b[h,g] then inc(w);
if w<>64 then prov(b) else begin
r:=1;
while r<=p do begin
for i1:=1 to 8 do begin
for i2:=1 to 8 do read(f,y[i1,i2]);
readln(f);
end;
for i1:=1 to 8 do begin
for i2:=1 to 8 do write(f1,y[i1,i2],' ');
writeln(f1);
end;
writeln(f1);
readln(f);
inc(r);
end;
end;
for i1:=0 to 7 do begin
for i2:=0 to 7 do write(f2,a[i1,i2],' ');
writeln(f2);
end;
writeln(f2);
reset(f1);
rewrite(f);
d:=1;
while d<=p do begin
for i1:=1 to 8 do begin
for i2:=1 to 8 do read(f1,y[i1,i2]);
readln(f1);
end;
for i1:=1 to 8 do begin
for i2:=1 to 8 do write(f,y[i1,i2],' ');
writeln(f);
end;
writeln(f);
readln(f1);
inc(d);
end;
{readln(f1);
writeln(f);
inc(n);
end;}
close(f);
close(f1);
close(f2);
writeln('ok');
readln;
end.
2. Текст программы поиска колец, имеющих аддитивную группу 
.
Программа 1. Нахождение всех базисов данной аддитивной группы:
program basis;
uses crt;
type mas=array [1..3] of integer;
var a,b,c,d,e,f,g:char;
i,j,n,k:integer;
x:mas;
f1,f2:text;
begin
assign(f2,'a:\28.txt');
reset(f2);
assign(f1,'a:\new168.txt');
rewrite(f1);
n:=1;
while n<=28 do begin
for i:=1 to 3 do read(f2,x[i]);
write(f1,x[1],' ',x[2],' ',x[3]);
writeln(f1);
write(f1,x[1],' ',x[3],' ',x[2]);
writeln(f1);
write(f1,x[3],' ',x[1],' ',x[3]);
writeln(f1);
write(f1,x[3],' ',x[2],' ',x[1]);
writeln(f1);
write(f1,x[2],' ',x[3],' ',x[1]);
writeln(f1);
write(f1,x[2],' ',x[1],' ',x[3]);
writeln(f1);
readln(f2);
writeln(f1);
inc(n);
end;
close(f2);
close(f1);
writeln('ok');
readln;
end.
Программа 2. Формирование полугрупп, с учетом ассоциативности:
program assZ2;
uses crt;
type mas=array [0..7,0..7] of integer;
var a,a1:mas;
i1,i2,i3,i4,i5,i6,i7,i8,i9,k1,k2,k3,r:integer;
n:real;
f,f1:text;
procedure sc(var a1:mas);
var i,j:integer;
begin
assign(f1,'a:\z2summ.txt');
reset(f1);
for i:=0 to 7 do begin
for j:=0 to 7 do read(f1,a1[i,j]);
readln(f1);
end;
close(f1);
end;
procedure zap(i1,i2,i3,i4,i5,i6,i7,i8,i9:integer;a1:mas; var r:integer);
var i,j,t1,t2,t3,t4,q:integer;
begin
q:=0;
for t1:=0 to 7 do a[0,t1]:=0;
for t2:=1 to 7 do a[t2,0]:=0;
a[1,1]:=i1; a[1,2]:=i2; a[1,3]:=i3;
a[2,1]:=i4; a[2,2]:=i5; a[2,3]:=i6;
a[3,1]:=i7; a[3,2]:=i8; a[3,3]:=i9;
for t3:=1 to 3 do begin
a[t3,4]:=a1[a[t3,1],a[t3,2]];
a[t3,5]:=a1[a[t3,1],a[t3,3]];
a[t3,6]:=a1[a[t3,2],a[t3,3]];
a[t3,7]:=a1[a[t3,1],a[t3,6]];
end;
for t4:=1 to 7 do begin
a[4,t4]:=a1[a[1,t4],a[2,t4]];
a[5,t4]:=a1[a[1,t4],a[3,t4]];
a[6,t4]:=a1[a[2,t4],a[3,t4]];
a[7,t4]:=a1[a[1,t4],a[6,t4]];
end;
{for i:=0 to 7 do begin
for j:=0 to 7 do write(a[i,j],' ');
writeln;
end;}
for k1:=0 to 7 do
for k2:=0 to 7 do
for k3:=0 to 7 do
if a[k1,a[k2,k3]]<>a[a[k1,k2],k3] then q:=1;
{writeln(q);}
if q=0 then begin
inc(r);
{for i:=0 to 2 do begin
for j:=0 to 2 do write(f,a[i,j],' ');
writeln(f);
end;
writeln(f);
end;
q:=0;}
end;
end;
begin
clrscr;
r:=0;
sc(a1);
assign(f,'a:\z2.txt');
rewrite(f);
{ i1:=6;i2:=6;i3:=6;i4:=6;i5:=6;i6:=6;i7:=6;i8:=6;i9:=6;}
for i1:=0 to 7 do
for i2:=0 to 7 do
for i3:=0 to 7 do
for i4:=0 to 7 do
for i5:=0 to 7 do
for i6:=0 to 7 do
for i7:=0 to 7 do
for i8:=0 to 7 do
for i9:=0 to 7 do begin
zap(i1,i2,i3,i4,i5,i6,i7,i8,i9,a1,r);
writeln(r);
end;
writeln(' ',r);
close(f);
writeln('ok');
readln;
end.
Программа 3. Нахождение изоморфных колец:
program izomorf;
uses crt;
const p=1;
type mas=array [0..7,0..7] of integer;
mas1=array [1..3] of integer;
var a,b,a1:mas;
y:mas1;
n,j1,i2,t,h,g,d:integer;
f,f1,f2,f3,f8:text;
procedure basis(var y:mas1);
var i:integer;
begin
for i:=1 to 3 do read(f2,y[i]);
readln(f2);
end;
procedure sc(var a:mas);
var i,j:integer;
begin
reset(f1);
for i:=0 to 7 do begin
for j:=0 to 7 do read(f1,a[i,j]);
readln(f1);
end;
end;
{procedure glav(b:mas);
var q,i,j,x,k:integer;
begin
x:=0;k:=1;
reset(f1);
rewrite(f8);
while n<=t do begin
q:=0;
for i:=0 to 7 do begin
for j:=0 to 7 do read(f1,a1[i,j]);
readln(f1);
end;
for i:=0 to 7 do
for j:=0 to 7 do
if a1[i,j]=b[i,j] then inc(q);
if q=64 then begin inc(h);
x:=1;
end
else
for i:=0 to 7 do begin
for j:=0 to 7 do write(f8,a1[i,j],' ');
writeln(f8);
end;
inc(n);
readln(f1);
writeln(f8);
end;
if x=1 then t:=t-2 else t:=t-1;
reset(f8);
rewrite(f1);
while k<=t do begin
for i:=0 to 7 do begin
for j:=0 to 7 do read(f8,a1[i,j]);
readln(f8);
end;
for i:=0 to 7 do begin
for j:=0 to 7 do write(f1,a1[i,j],' ');
writeln(f1);
end;
inc(k);
end;
end;}
procedure newb(y:mas1;a:mas; var b:mas);
var i,j,z1,z2,z3:integer;
begin
for i:=0 to 7 do begin
z1:=a[1,i]; a[1,i]:=a[y[1],i]; a[y[1],i]:=z1;
z2:=a[2,i]; a[2,i]:=a[y[2],i]; a[y[2],i]:=z2;
z3:=a[3,i]; a[3,i]:=a[y[3],i]; a[y[3],i]:=z3;
end;
for j:=0 to 7 do begin
z1:=a[j,1]; a[j,1]:=a[j,y[1]]; a[j,y[1]]:=z1;
z2:=a[j,2]; a[j,2]:=a[j,y[2]]; a[j,y[2]]:=z2;
z3:=a[j,3]; a[j,3]:=a[j,y[3]]; a[j,y[3]]:=z3;
end;
for i:=0 to 7 do
for j:=0 to 7 do begin
if a[i,j]=1 then a[i,j]:=-1;
if a[i,j]=y[1] then a[i,j]:=-y[1];
end;
for i:=0 to 7 do
for j:=0 to 7 do begin
if a[i,j]=-1 then a[i,j]:=y[1];
if a[i,j]=-y[1] then a[i,j]:=1;
end;
for i:=0 to 7 do
for j:=0 to 7 do begin
if a[i,j]=2 then a[i,j]:=-2;
if a[i,j]=y[2] then a[i,j]:=-y[2];
end;
for i:=0 to 7 do
for j:=0 to 7 do begin
if a[i,j]=-2 then a[i,j]:=y[2];
if a[i,j]=-y[2] then a[i,j]:=2;
end;
for i:=0 to 7 do
for j:=0 to 7 do begin
if a[i,j]=3 then a[i,j]:=-3;
if a[i,j]=y[3] then a[i,j]:=-y[3];
end;
for i:=0 to 7 do
for j:=0 to 7 do begin
if a[i,j]=-3 then a[i,j]:=y[3];
if a[i,j]=-y[3] then a[i,j]:=3;
end;
for i:=0 to 7 do
for j:=0 to 7 do b[i,j]:=a[i,j];
{glav(b);}
for i:=0 to 7 do begin
for j:=0 to 7 do write(f,b[i,j],' ');
writeln(f);
end;
writeln(f);
end;
procedure new5(b,a:mas);
var i,j,z1,z2,i1:integer;
begin
for i:=1 to 5 do begin
if i mod 2<>0 then begin
for j:=0 to 7 do begin
z1:=b[2,j]; b[2,j]:=b[3,j]; b[3,j]:=z1;
end;
for j:=0 to 7 do begin
z2:=b[j,2]; b[j,2]:=b[j,3]; b[j,3]:=z2;
end;
for i1:=0 to 7 do
for j:=0 to 7 do begin
if b[i1,j]=3 then b[i1,j]:=-3;
if b[i1,j]=2 then b[i1,j]:=-2;
end;
for i1:=0 to 7 do
for j:=0 to 7 do begin
if b[i1,j]=-3 then b[i1,j]:=2;
if b[i1,j]=-2 then b[i1,j]:=3;
end;
{glav(b);}
for i1:=0 to 7 do begin
for j:=0 to 7 do write(f,b[i1,j],' ');
writeln(f);
end;
writeln(f);
end;
if i mod 2=0 then begin
for j:=0 to 7 do begin
z1:=b[1,j]; b[1,j]:=b[2,j]; b[2,j]:=z1;
end;
for j:=0 to 7 do begin
z2:=b[j,1]; b[j,1]:=b[j,2]; b[j,2]:=z2;
end;
for i1:=0 to 7 do
for j:=0 to 7 do begin
if b[i1,j]=1 then b[i1,j]:=-1;
if b[i1,j]=2 then b[i1,j]:=-2;
end;
for i1:=0 to 7 do
for j:=0 to 7 do begin
if b[i1,j]=-1 then b[i1,j]:=2;
if b[i1,j]=-2 then b[i1,j]:=1;
end;
{glav(b);}
for i1:=0 to 7 do begin
for j:=0 to 7 do write(f,b[i1,j],' ');
writeln(f);
end;
writeln(f);
end;
end;
end;
begin
clrscr;
assign(f1,'a:\vse_z21.txt');
reset(f1);
assign(f2,'a:\28.txt');
reset(f2);
assign(f,'a:\xx.txt');
rewrite(f);
{assign(f3,'a:\ist.txt');
rewrite(f3);
assign(f8,'a:\pom.txt');
rewrite(f8);}sc(a);
for g:=1 to 28 do begin
basis(y);{d:=1;g:=0;}
{while d<=t do begin}
{inc(g);writeln(g);}
{for j1:=0 to 7 do begin
for i2:=0 to 7 do write(f3,a[j1,i2]);
writeln(f3);
end;
writeln(f3);}
newb(y,a,b);
{n:=1;}
new5(b,a);
{inc(d);}
end;
close(f);
close(f1);
close(f2);
{close(f3);}
{close(f8);}
{ writeln(h); }
writeln('ok');
readln;
end.