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Projective Prime Ideals and Localisation in PI‐Rings
Author(s) -
Chatters A. W.,
Hajarnavis C. R.,
Lissaman R. M.
Publication year - 2001
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1017/s0024610701002125
Subject(s) - mathematics , associated prime , noetherian ring , ideal (ethics) , prime ideal , idempotence , noetherian , prime (order theory) , minimal ideal , combinatorics , radical of a ring , element (criminal law) , discrete mathematics , maximal ideal , pure mathematics , principal ideal ring , commutative ring , algebra over a field , finitely generated abelian group , commutative property , law , political science
The results here generalise [ 2 , Proposition 4.3] and [ 9 , Theorem 5.11]. We shall prove the following. THEOREM A. Let R be a Noetherian PI‐ring. Let P be a non‐idempotent prime ideal of R such that P R is projective. Then P is left localisable and R P is a prime principal left and right ideal ring. We also have the following theorem. THEOREM B. Let R be a Noetherian PI‐ring. Let M be a non‐idempotent maximal ideal of R such that M R is projective. Then M has the left AR‐property and M contains a right regular element of R.