Regular ideal

In mathematics, especially ring theory, a regular ideal can refer to multiple concepts.

In operator theory, a right ideal ${\displaystyle {\mathfrak {i}}}$ in a (possibly) non-unital ring A is said to be regular (or modular) if there exists an element e in A such that ${\displaystyle ex-x\in {\mathfrak {i}}}$ for every ${\displaystyle x\in A}$.^[1]

In commutative algebra a regular ideal refers to an ideal containing a non-zero divisor.^[2][3] This article will use "regular element ideal" to help distinguish this type of ideal.

A two-sided ideal ${\displaystyle {\mathfrak {i}}}$ of a ring R can also be called a (von Neumann) regular ideal if for each element x of ${\displaystyle {\mathfrak {i}}}$ there exists a y in ${\displaystyle {\mathfrak {i}}}$ such that xyx=x.^[4][5]

Finally, regular ideal has been used to refer to an ideal J of a ring R such that the quotient ring R/J is von Neumann regular ring.^[6] This article will use "quotient von Neumann regular" to refer to this type of regular ideal.

Since the adjective regular has been overloaded, this article adopts the alternative adjectives modular, regular element, von Neumann regular, and quotient von Neumann regular to distinguish between concepts.