# 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 $I$:

$$re\equiv r\phantom{\rule{veryverythickmathspace}{0ex}}(modI).$$ |

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 |