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\eq 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 $a_{i}\\in\\mathfrak{a}_{i}\\,\\,\\,\\forall i$ .