Local Rings, Semilocal Rings, and Idempotents
In commutative algebra, a local ring is defined to be a (nonzero) ring which has a unique maximal ideal. This definition generalizes readily to arbitrary rings: a (nonzero) ring A is said to be local if A has a unique maximal left ideal. This definition turns out to be left-right symmetric, and is equivalent to the condition that A/rad A be a division ring.
KeywordsLocal Ring Direct Summand Division Ring Valuation Ring Stable Range
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