Lakhasly

3 Sum and direct sum of Ideals In this section we discuss the concept of the sum of ideals (right ideals, left ideals) in a ring R. Definition...,R,, be afamily of rings, and let R = R1 X R2 X" X R,, be their direct product. Let R? E R,). Then R -- is a direct sum of ideals R?, and R? R, as rings. On the other hand, fR = (R)17...IA,, adirect sum of ideals ofR, then R A1 X A2 X X A,,, the direct product oftheA,'s considered as rings on their own right. Proof Clearly, R R a R. Let xERrfl j*, Then x=(0,0,...,a,,0,...,0)(a1,a2,...,a,_1,0,a,+1,...,a,J. This gives a, = 0 and, hence, x =0. Therefore, 198 Ideals and bomoinorpldsns For the second part we note that ifxE R, then x can be uniquely expressed asa1+a2++ a,a,EA,, by Because and onto. Also, if x,y isdirect,f is well defined. It is also clearthatf is both 1-1 (R)17..IA,, thenf(x + y) =f(x) +f(y). To show we need to note that if a1 + since a,b, E A, fl 4 (0). This remark then immediately yields that f(xy) =f(x)f(y). Hence,f is an isomorphism. 0 Thedirect sumR -- (R) 17..1A,is also call ed the (internal) direct swn of idealsA,inR, (external) direct sum of the family of rings A,, i -- 1,2,...,n. In the tatter case the notation A1 (R)A2 0... 0 A,, is also frequently used. The context will make clear the sense in which the term "direct sum" is used.If A,,A2 A,, are right ideals in a ring R, then S(a1 +a2+ +a,,Ia,EA,, i= 1,2 n) is the of right ideals A1,A2 A,, Proof It is clear that (a, + a2 + + a,ja, E A,, i= 1,...,n) isa right ideal in R. Also, if a,EA,, then a1=a1 in S, and, hence, A, C S. Similarly, each A,, i 2,...,n, is contained in S. Further, if Tis any right ideal in R containing each A,, then obviouslyT S. Thus, S is the intersection of all the right ideals in R containing each A,.LeIA, ,A2,....A,, beafa,nilyofright ideals in a ringR.is a minimal right ideal, then I is generated by any nonzero element of!.3.2 Theorem.LetA1 right (or left) ideals inaringR.

3 Sum and direct sum of Ideals
In this section we discuss the concept of the sum of ideals (right ideals, left
ideals) in a ring R.
Definition. LeIA, ,A2,....A,, beafa,nilyofright ideals in a ringR. Then the
smallest right ideal of R containing each A,, 1 i
n (that is, the inter-
section of all the right ideals in R containing each A,), is called the sum of
A,,A2
3.1 Theorem. If A,,A2 A,, are right ideals in a ring R, then
S(a1 +a2+ +a,,Ia,EA,, i= 1,2 n) is the of right ideals
A1,A2 A,,
Proof It is clear that
(a, + a2 + + a,ja, E A,, i= 1,...,n)
isa right ideal in R. Also, if a,EA,, then a1=a1 in S, and,
hence, A, C S. Similarly, each A,, i 2,...,n, is contained in S. Further, if
Tis any right ideal in R containing each A,, then obviouslyT S. Thus, S
is the intersection of all the right ideals in R containing each A,. 0
Notation. The sum of right (or left) ideals A1 ,A2,...,A,, in a ring R is
denoted by A, + A2 + + A,,. From the definition of the sum it is clear
that the order of A,'s in A, + A2 + + A,, is immaterial. We write
A A = of right (or left) ideals in a ring R is called a
direct sum (leach element a E A is uniquely expressible in the for,n
17_,a,. a, C A,, I I n. IfthesumA = LI,is a direct sum, we write it as
A=A,®A2®
Note. One can similarly define the sum and direct sum of an infinite
family of right (left) ideals in a ring R. Although no extra effort is needed
to talk about this, we prefer to postpone it to Chapter 14, where we
discuss sum and direct sum of a more general family.
3.2 Theorem.LetA1 right (or left) ideals inaringR. Then
the following are equivalent:
Sum and direct sum of Ideals
(I) A — is a direct sum.
(ii) ff0— E7_1a,, then a,=0, i= l,2....,n.
(iii) fl (0), i 1,2,...,n.
Proof (I) : (ii) Follows from definition of direct sum.
(ii) (iii) Let x E fl Then
x— a1+ a,_1+ a,, A,.
Thus,
Then by (ii) we get x =0.
Let a=a1+a2+'"+a,, and a=b1+b2+"'+b,,,
where i='l,2,...,n. Then O=(a1—b1)+(a2—b2)+"+
(a, — b,). This gives
a1 —b1 EA1
Hence, a1 — b,. Similarly, a2 b2,...,a,, = b,,. Hence, A direct sum. 0
is a
3.3 Theorem. Let 1,R2,. ..,R,, be afamily of rings, and let R = R1 X
R2 X" X R,, be their direct product. Let R? E
R,). Then R — is a direct sum of ideals R?, and R? R, as
rings. On the other hand, fR = ®17...IA,, adirect sum of ideals ofR, then
R A1 X A2 X X A,,, the direct product oftheA,'s considered as rings
on their own right.
Proof Clearly, R
R a R. Let
xERrfl
j*,
Then
x=(0,0,...,a,,0,...,0)(a1,a2,...,a,_1,0,a,+1,...,a,J.
This gives a, = 0 and, hence, x =0. Therefore,
198 Ideals and bomoinorpldsns
For the second part we note that ifxE R, then x can be uniquely expressed asa1+a2++ a,a,EA,,
by
Because and onto. Also, if x,y isdirect,f is well defined. It is also clearthatf is both 1-1
®17..IA,, thenf(x + y) =f(x) +f(y). To show
we need to note that if a1 +
since a,b, E A, fl 4 (0). This remark then immediately yields that
f(xy) =f(x)f(y). Hence,f is an isomorphism. 0
Thedirect sumR — ® 17..1A,is also call ed the (internal) direct swn of
idealsA,inR,
(external) direct sum of the family of rings A,, i — 1,2,...,n. In the tatter
case the notation A1 ®A2 0... 0 A,, is also frequently used. The context
will make clear the sense in which the term "direct sum" is used.
Definition. A right (left) ideal fin a ring R is coiled minimal jf(i) I # (0),
and (ii) if I is a nonzero right (left) ideal of R contained in I, then I — I.
It is clear that if! is a minimal right ideal, then I is generated by any
nonzero element of!. Indeed, if! is a right ideal of R with the property that
each nonzero element generates!, then! is minimal. To see this, let Jbe a
right ideal of R such that 0 # IC I. Suppose 0 # a E J. By assumption,
I (a), C I, so I — I.
If R is a division ring, then R itself is a minimal right ideal as well as a
minimal left ideal. A nontrivial example of a minimal right ideal is