Star operation on orders in simple Artinian rings

TL;DR

This paper explores the application of $ u$-star operations to define and analyze right Pr"ufer $ u$-multiplication orders in simple Artinian rings, extending concepts from commutative to non-commutative algebra.

Contribution

It introduces the notion of right Pr"ufer $ u$-multiplication orders and proves their stability under overrings, advancing non-commutative multiplicative ideal theory.

Findings

Overrings of right Pr"ufer $ u$-multiplication orders are also right Pr"ufer $ u$-multiplication orders.

The paper establishes a non-commutative analogue of Pr"ufer $ u$-multiplication domains.

It demonstrates the utility of $ u$-star operations in non-commutative ring theory.

Abstract

Star operations are an important tool in multiplicative ideal theory. In this paper we apply a special type of star operation, known as $\nu$-operation, to define the notion of right Pr\"ufer $\nu$-multiplication order. The latter may be viewed as a natural non-commutative version of Pr\"ufer $\nu$-multiplication domain. As one of our main results, we establish that an overring of a right Pr\"ufer $\nu$-multiplication order is again a right Pr\"ufer $\nu$-multiplication order.