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