# R-Modules

• Dec 29th 2011, 12:04 AM
jcir2826
R-Modules
Let M = S_1 coproduct S_2 .... coproduct S_n. be a direct sum of R-modules. If T_i is a subset of S_i for all i prove that

(S_1 coproduct .... coproduct S_n)/(T_1 coproduct ....T_n) is isomorphic to

(S_1/T_1) co product (S_2/T_2) ---- coproduct (S_n/T_n).
• Dec 30th 2011, 11:15 AM
Drexel28
Re: R-Modules
Quote:

Originally Posted by jcir2826
Let M = S_1 coproduct S_2 .... coproduct S_n. be a direct sum of R-modules. If T_i is a subset of S_i for all i prove that

(S_1 coproduct .... coproduct S_n)/(T_1 coproduct ....T_n) is isomorphic to

(S_1/T_1) co product (S_2/T_2) ---- coproduct (S_n/T_n).

Just do what's natural, define $f:S_1\oplus\cdots\oplus S_n\to (S_1/T_1)\oplus\cdots\oplus(S_n/T_n)$ by $(s_1,\cdots,s_n)\mapsto (s_1+T_1,\cdots,s_n+T_n)$. Is this map surjective? What is the kernel of this map? Why do you care?