prime ring

Example Sentences

• "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."