Which of the following pairs R and M are not an example of a module? a. M is a subring of a ring R b. R, any commutative ring, and I an ideal of R. R=F a field; M an F-vector space d. R = Z, the ring of integers; M, any abelian group C.

Let R be an abelian group. View R as a ring by declaring the multiplication operation is ab = 0 for all a, b = R. Prove that every subgroup S ≤ R is actually an ideal.

Let R be a ring and M a left R-module. Show that, for any idempotent e ER End(M) We have ker(e) = im(Id-e)

Lot R bx unity commutative ring with (1) Prove that if a ER, then a R={arIVER} а 1) an ideal of R (b) is aR a maximal ideal. a