(a) Find all the units for each of the following rings. Justify your answers briefly.
i. Z15.
ii. Z11.
iii. Z × Q × Z3.
(b) How many units are in M2(Z3), the ring of all 2 × 2 matrices with entries in Z3?
Briefly justify your answer.
(c) Let R be a ring with multiplicative identity and U be the set of all units in R. Prove that U is a group under multiplication.
(d) Prove that for any ring R, no element a ∈ R is both a unit and a zero divisor.
(e) Show that if ab = 1 in a ring R with multiplicative identity 1 and no zero divisors, then ba = 1 also (even if the ring is not a commutative ring).