internal direct sum of ideals

Let $R$ be a ring and $\mathfrak{a}_{1}$ , $\mathfrak{a}_{2}$ , …, $\mathfrak{a}_{n}$ its ideals (left, right or two-sided). We say that $R$ is the internal direct sum of these ideals, denoted by

R=\mathfrak{a}_{1}\oplus\mathfrak{a}_{2}\oplus\cdots\oplus\mathfrak{a}_{n},

if both of the following conditions are true:

R=\mathfrak{a}_{1}+\mathfrak{a}_{2}+\cdots+\mathfrak{a}_{n},

\mathfrak{a}_{i}\cap\sum_{j\neq i}\mathfrak{a}_{j}=\{0\}\quad\forall i.

Theorem.

If $\mathfrak{a}_{1}$ , $\mathfrak{a}_{2}$ , …, $\mathfrak{a}_{n}$ are ideals of the ring $R$ , then the following two statements are equivalent:

•

$R=\mathfrak{a}_{1}\oplus\mathfrak{a}_{2}\oplus\cdots\oplus\mathfrak{a}_{n}$ .
•

Every element $r$ of $R$ has a unique expression
$r=a_{1}\!+\!a_{2}\!+\cdots+\!a_{n}$ with $a_{i}\in\mathfrak{a}_{i}\,\,\,\forall i$ .

Title	internal direct sum of ideals
Canonical name	InternalDirectSumOfIdeals
Date of creation	2013-03-22 14:49:32
Last modified on	2013-03-22 14:49:32
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	7
Author	Mathprof (13753)
Entry type	Theorem
Classification	msc 16D25
Classification	msc 11N80
Classification	msc 13A15
Defines	internal direct sum of ideals