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Another rings with isomorphism between its multiplicative and additive groups

Source: Romanian District Olympiad 2016, Grade XII, Problem 1

October 5, 2018
superior algebraRing Theoryabstract algebragroup theoryFTA

Problem Statement

A ring A A has property (P), if A A is finite and there exists ({0}≠R,+)≤(A,+) (\{ 0\}\neq R,+)\le (A,+) such that (U(A),⋅)≅(R,+). (U(A),\cdot )\cong (R,+) . Show that:
a) If a ring has property (P), then, the number of its elements is even. b) There are infinitely many rings of distinct order that have property (P).
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