 
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.
