Summary: Rings and Algebras Problem set #5. Oct. 13, 2011.
The ring R is a subdirect product of the rings Ri if R i Ri in such a way that each projection
R Ri is surjective. The ring is subdirectly irreducible, if and only if whenever R i Ri
as a subdirect product then at least one of the projections i : R Ri 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 non-zero ideals
of R is non-zero, i. e. R has a unique minimal (non-zero) ideal which is contained in each
c) (Birkhoff) Every ring is a subdirect product of subdirectly irreducible rings.
d) Zn 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 a2