$UA$-properties of modules over commutative Noetherian rings

A semigroup $(R,\cdot)$ is said to be a $UA$-ring if there exists a unique binary operation $+$ transforming $(R,\cdot,+)$ into a ring. An $R$-module $A$ is said to be a $UA$-module if it is not possible to change the addition of $A$ without changing the action of $R$ on $A$. In this paper we investigate topics that are related to the structure of $UA$-rings of endomorphisms and $UA$-modules over commutative Noetherian rings.