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 non-commutative 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 K-algebra generated by a countably infinite set, K x1, x2, . . . is isomorphic to a subalgebra of K x, y . 6. Let V be a vector space with basis {e1, . . . , en}. 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 = R1 ˇ ˇ ˇ Rn (i. e. R is a ring direct sum of some two-sided ideals). Determine the left, right and two-sided ideals of R. b) Let B1, . . . , Bn be left ideals (resp. two sided ideals) in the ring R. Show tha R is the Collections: Mathematics