Periodic Abelian groups with UA-rings of endomorphisms
A ring \((R,\cdot,+)\) is called a unique addition ring (UA-ring) if whenever the multiplicative semigroup \((R,\cdot)\) can be endowed with a binary operation \(+'\) such that \((R,\cdot,+')\) is a ring, then \(+\) and \(+'\) are the same binary operation. An Abelian group is called an End-UA-group if its endomorphism ring is a UA-ring. In the current paper, the author finds the End-UA groups in the class of torsion Abelian groups. Not surprisingly, this problem comes down to finding the primary UA-groups. The main results are: Theorem 1. Let \(G\) be a \(p\)-group with an unbounded basic subgroup. Then \(G\) is an End-UA-group. Theorem 2. Let \(G=B\oplus D\) where \(B\) is bounded and \(D\) is a nonzero divisible \(p\)-group. Then \(G\) is not an End-UA-group if and only if \(D=\mathbb{Z}_p(\infty)\). Theorem 3. Let \(G\) be a bounded \(p\)-group with \(G\not=\mathbb{Z}_2,\mathbb{Z}_3\). Then \(G\) is not an End-UA-group if and only if it contains an isolated direct summand. Prior to Theorem 3, the author had established that a finite cyclic \(p\)-group \(G\) is an End-UA-group if and only if \(G=\mathbb{Z}_2\) or \(G=\mathbb{Z}_3\). The author also claims to have a description of torsion modules over commutative Dedekind rings whose endomorphism rings are UA-rings, but omits the details.
scientific article; zbMATH DE number 1927833