modular ideal

Let $R$ be a ring. A left ideal $I$ of $R$ is said to be modular if there is an $e\in R$ such that $re-r\in I$ for all $r\in R$ . In other words, $e$ acts as a right identity element modulo $I$ :

re\equiv r\pmod{I}.

A right modular ideal is defined similarly, with $e$ be a left identity modulo $I$ .

Remark. If an ideal $I$ is modular both as a left ideal as well as a right ideal in $R$ , then $R/I$ is a unital ring. Furthermore, every (left, right, two-sided) ideal in a unital ring is modular, implying that the notion of modular ideals is only interesting in rings without $1$ .

References

1 P. M. Cohn, Further Algebra and Applications, Springer (2003).

Title	modular ideal
Canonical name	ModularIdeal
Date of creation	2013-03-22 17:31:47
Last modified on	2013-03-22 17:31:47
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	7
Author	CWoo (3771)
Entry type	Definition
Classification	msc 16D25