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 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 non­zero ideal. 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; Collections: Mathematics