Meaning of simple ring | Babel Free

Definitions

A ring that contains no nontrivial ideals (i.e., no (two-sided) ideals other than the zero ideal and the ring itself).

Equivalents

Examples

“Theorem 2. A semi-simple ring⟳ #92;mathfrak#123;A#125; satisfying the minimum condition can be decomposed in only one way as a direct⟳ sum of ideals which are simple rings.”

“By theorem 7.7.1 any field is a simple ring⟳.”

“A field is clearly a simple ring⟳. Indeed, a commutative simple ring⟳ with unity must be a field (Problem 1, Section 1). An example of a noncommutative simple ring⟳ is Fₙ, the n × n matrix ring⟳ over a field F, n > 1.”

“Since M#95;n(D) has no nonzero proper two-sided ideals, it follows from the above discussion that M#95;n(D) is a left simple ring⟳. We shall see⟳ that every simple ring⟳ is isomorphic to M#95;n(D) for some n, and for some division ring⟳ D.”

CEFR level