Prime and Primitive Rings
The notion of prime ideals in commutative rings extends naturally to non-commutative rings. One takes the “same” definition, but replaces the use of elements by ideals of the ring. The Zorn’s Lemma construction of prime ideals disjoint from a multiplicative set in the commutative setting finds a natural generalization, if we just replace the multiplicative set with an “m-system”: cf. FC-(10.5). (A nonempty set S ⊆ R is called an m-system if, for any a, b ∊ S, arb ∊ S for some r ∊ R.)
KeywordsPrime Ideal Left Ideal Prime Ring Division Ring Density Theorem
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