11.2 The Wedderburn-Artin Theorem
1. The 
-modules are the (additive) abelian groups, so the simple ones are the simple abelian groups. These are the prime cycles 
(in additive notation) where p is a prime. Hence the semisimple 
-modules are the the direct sums of copies of these 
for various primes p. The homogeneous semisimple 
-modules are the direct sums of copies of 
for a fixed prime p.
-modules are the (additive) abelian groups, so the simple ones are the simple abelian groups. These are the prime cycles (in additive notation) where p is a prime. Hence the semisimple -modules are the the direct sums of copies of these for various primes p. The homogeneous semisimple -modules are the direct sums of copies of for a fixed prime p.
2. a. Since RR is complemented by Theorem 1, we have R = L ⊕ M for some left ideal M. Write 1 = e + f where e 
L and f M. Hence Re ⊆ L, and we claim this is equality. If x L then x − xe = xf L ∩ M = 0 because ...
L and f M. Hence Re ⊆ L, and we claim this is equality. If x L then x − xe = xf L ∩ M = 0 because ...Get Introduction to Abstract Algebra, Solutions Manual, 4th Edition now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
