Rings and Algebras Problem set #1 Sept. 15, 2011. 1. Prove that the following are equivalent for a ring R Summary: Rings and Algebras Problem set #1 Sept. 15, 2011. 1. Prove that the following are equivalent for a ring R: a) R has an identity element; b) whenever R # S for a ring S (possibly without an identity), then R is a ring direct summand of S (i. e. there is an ideal T # S such that R # T = {0} and R + T = S). 2. Give an example of a non­commutative ring R for which the multiplicative group of invertible elements is commutative. 3. Let a, b be elements of a ring R. Show that if 1 - ab is left invertible then 1 - ba is also left invertible. 4. Show that if an element a # R has at least 2 right inverses then it has infinitely many right inverses. 5. Show that the free K­algebra generated by a countably infinite set, K#x 1 , x 2 , . . .# is isomorphic to a subalgebra of K#x, y#. 6. Let V be a vector space with basis {e 1 , . . . , e n }. Give a basis for the exterior algebra # (V ). 7. a) Let k be an arbitrary ring. Characterize the set of invertible elements in k [[ x ]], the ring of formal power series over k. b) Show that if k is a field, then the ring of formal Laurent series, k (( x )) is also a field. 8. a) Let R = R 1 # ˇ ˇ ˇ # R n (i. e. R is a ring direct sum of some two­sided ideals). Determine the left, right and two­sided ideals of R. Collections: Mathematics