Meaning of prime ring | Babel Free

Definitions

1. 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.
2. Synonym of prime subring.
3. A ring which is equal to its own prime subring.
   uncommon

Examples

“A ring⟳ is called a prime ring⟳ if the product of nonzero ideals in it remains nonzero. It is obvious that a prime ring⟳ is necessarily semi-prime.”

“The ring⟳ R is said to be prime if for all nonzero ideals A, B of R we have⟳ AB≠0. An ideal P of R is called a prime ideal if R/P is a prime ring⟳. Prime rings and prime ideals are important building blocks in noncommutative ring⟳ theory.”

“The so-called extended centroid of a prime ring⟳, i.e., a field defined as the center of the Martindale ring⟳ of quotients, will enable us to extend⟳ a part of the theory of central simple algebras to general prime rings.”

“Moreover, the image of φ_A is the smallest subring of A, in the sense⟳ that it is contained in any subring of A, and it is called the prime ring⟳ of A.”

“The image of ℤ or ℤ/mℤ, respectively, in R as described in the above proposition obviously consists of all sums n · 1 in R, where n ∈ ℤ. It is also called the prime ring⟳ of R. R itself is called a prime ring⟳ if it equals its own⟳ prime ring⟳. If p is a prime number⟳, then the field ℤ/pℤ is a also called the prime field of characteristic p.”

“Theorem 3.6.3. If R is a prime ring⟳ of characteristic zero then R is isomorphic to ℤ. If R is a prime ring⟳ of characteristic n then R is isomorphic to ℤₙ.”

CEFR level