Abstract Algebra Answers

Construct an artinian ring R in which soc(R_R) is not equal soc(RR).

For any left artinian ring R with Jacobson radical J, show that
soc(R_R) = {r ∈ R : Jr = 0} and soc(RR) = {r ∈ R : rJ = 0}.

Give proof for the fact that if R is a simple ring which has a minimal left ideal, then R is a semisimple ring.

Show that for any ring R, soc(R) is an ideal of R.

Show that soc(M) ⊆ {m ∈ M : (rad R) • m = 0}, with equality if R/rad R is an artinian ring.

Let R be a ring in which all descending chains Ra ⊇ Ra2 ⊇ Ra3 ⊇ • • • (for a ∈ R) stabilize. Show that R is Dedekind-finite, and every non right-0-divisor in R is a unit.

Let ϕ : R → S be a ring homomorphism such that S is finitely generated when it is viewed as a left R-module via ϕ. If, over R, all finitely generated left modules are hopfian (resp. cohopfian), show that the same property holds over S.

Show that the left regular module R is cohopfian iff every non right-0-divisor in R is a unit. In this case, show that R is also hopfian

Show that any artinian module M is cohopfian.

TWO ATHLETES PETER AND JOHN HAVE THE PROBABILITY OF 1/3 AND 3/4 RESPECTIVELY TO QUALIFY FOR THE FINALS OF A HIGH JUMP.IF THEIR ATTEMPTS ARE INDEPENDENT,DETERMINE THE PROBABILITY THAT,
BOTH WILL QUALIFY FOR THE FUNDS