Semidistributive Modules and Rings pp 101-132 | Cite as

# Rings of quotients

## Abstract

Let *T* be a set of elements in a ring *A*. The set *T* is *right permutable* if for any *a* ∈ *A* and *t* ∈ *T*, there exist *b* ∈ *A*, *u* ∈ *T* such that *au* = *tb*. A *multiplicative* set in a ring *A* is any subset *T* of *A* such that 1 ∈ *T*,0 ∉ *T* and *T* is closed under multiplication. A *completely prime* ideal in a ring *A* is any proper ideal *B* such that *A*\*B* is a multiplicative set (i.e. *A*/*B*is a domain). A *minimal prime* ideal (resp. *minimal completely prime* ideal) in a ring *A* is any prime (resp. completely prime) ideal *P* such that *P* contains no properly any other prime ideal (resp. completely prime ideal) of *A*. Let *I* be any proper ideal of a ring *A*. The set of all elements *a* ∈ *A* such that *a* + *I* is a regular element of *A*/*I* is denoted by *c*(*I*). In particular, *c*(0) is the set of all regular elements of *A*.

## Keywords

Prime Ideal Division Ring Regular Element Semiprime Ring Minimal Prime Ideal## Preview

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