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Injectivity of aA, Aa, for A ring

Source: Romania National Olympiad 2014, Grade XII, Problem 1

March 3, 2019
functionRing Theory

Problem Statement

For a ring A, A, and an element a a of it, define sa,da:A⟶A,sa(x)=ax,da=xa. s_a,d_a:A\longrightarrow A, s_a(x)=ax,d_a=xa.
a) Prove that if A A is finite, then sa s_a is injective if and only if da d_a is injective. b) Give example of a ring which has an element b b for which sb s_b is injective and db d_b is not, or, conversely, sb s_b is not injective, but db d_b is.
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