Prime ring
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English
Noun
- Template:Senseid Template:Lb Any nonzero ring R such that for any two (two-sided) ideals P and Q in R, the product PQ = 0 (the zero ideal) if and only if P = 0 or Q = 0.
- Template:Lb Template:Synonym of
- Template:Senseid Template:Lb A ring which is equal to its own prime subring.
Usage notes
The following conditions are equivalent to R being a prime ring (Template:Senseno):
- for arbitrary a, b ∈ R, if arb = 0 for all r ∈ R (i.e., if aRb = 0) then either a = 0 or b = 0;
- the zero ideal is a prime ideal in R.
Template:Senseno and Template:Senseno are not equivalent; for example, is a prime ring in the sense of Template:Senseno, but not in the sense of Template:Senseno.