Summary: Rings and Algebras Problem set #5. Oct. 13, 2011.
The ring R is a subdirect product of the rings R i if R # # i R i in such a way that each projection
R # R i is surjective. The ring is subdirectly irreducible, if and only if whenever R # # i R i
as a subdirect product then at least one of the projections # i : R # R i is injective. (Note that
both of these notions can be generalized to arbitrary algebraic structures.)
1. Show the following statements:
a) A ring R is semiprimitive if and only if R is a subdirect product of primitive rings.
b) A ring R is subdirectly irreducible if and only if the intersection of all the nonzero ideals
of R is nonzero, i. e. R has a unique minimal (nonzero) ideal which is contained in each
c) (Birkho#) Every ring is a subdirect product of subdirectly irreducible rings.
d) Z n is subdirectly irreducible if and only if n = p # for some prime p.
2. a) Show that if R is a prime ring with soc R #= 0 then R is subdirectly irreducible.
b) Show that a simple nonartinian ring is subdirectly irreducible prime ring with soc R = 0.
Give an example of such a ring.
3. Let R be subdirectly irreducible. Show that if J(R) = 0 then R is left primitive. Hence show
that a subdirectly irreducible ring is left primitive if and only if it is right primitive.
4. A ring R is called strongly (von Neumann) regular if for every a # R there exists an x # R
such that a 2 x = a. Show that if R is strongly regular then:
a) R is reduced, i. e. it does not contain nonzero nilpotent elements;