# 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 InternalDirectSumOfIdeals 2013-03-22 14:49:32 2013-03-22 14:49:32 Mathprof (13753) Mathprof (13753) 7 Mathprof (13753) Theorem msc 16D25 msc 11N80 msc 13A15 internal direct sum of ideals