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modular ideal (Definition)

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$:

$\displaystyle re\equiv r\pmod I.$
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$.

Bibliography

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



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Cross-references: unital ring, right ideal, ideal, left identity, right, right identity, modular, left ideal, ring
There is 1 reference to this entry.

This is version 4 of modular ideal, born on 2007-09-10, modified 2007-09-10.
Object id is 9926, canonical name is ModularIdeal.
Accessed 432 times total.

Classification:
AMS MSC: 16D25 (Associative rings and algebras :: Modules, bimodules and ideals :: Ideals)
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