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1997•Quaestiones MathematicaeRequires access

PRIME IDEALS IN RINGS WITH INVOLUTION

G. F. Blrkenmeier, N. J. Groenewald

Open publisher page 10 citations

Abstract

Let R be a ring with involution *. We show that if R is a *-prime ring which is not a prime ring, then R is “essentially” a direct product of two prime rings. Moreover, if P is a *-prime *-ideal of R, which is not a prime ideal of R, and X is minimal among prime ideals of R containing P, then P is a prime ideal of X, P = X ∩ X* and either: (1) P is essential in X and X is essential in R; or (2) for any relative complement C of P in X, then C* is a relative complement of X in R. Further characterizations of *-primeness are provided.

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What this paper is about

Let R be a ring with involution *. We show that if R is a *-prime ring which is not a prime ring, then R is “essentially” a direct product of two prime rings. Moreover, if P is a *-prime *-ideal of R, which is not a prime ideal of R, and X is minimal among prime ideals of R containing P, then P is a prime ideal of X, P = X ∩ X* and either: (1) P is essential in X and X is essential in R; or (2) for any relative complement C of P in X, then C* is a relative complement of X in R. Further characterizations of *-primeness are provided.

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Available abstract

Let R be a ring with involution *. We show that if R is a *-prime ring which is not a prime ring, then R is “essentially” a direct product of two prime rings. Moreover, if P is a *-prime *-ideal of R, which is not a prime ideal of R, and X is minimal among prime ideals of R containing P, then P is a prime ideal of X, P = X ∩ X* and either: (1) P is essential in X and X is essential in R; or (2) for any relative complement C of P in X, then C* is a relative complement of X in R. Further characterizations of *-primeness are provided.

Key concepts: Mathematics, Involution (esoterism), Associated prime, Prime (order theory), Prime ideal, Ideal (ethics), Prime ring, Complement (music)

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