Glasnik Matematicki, Vol. 40, No.1 (2005), 29-45.

THE θ-TRANSFER TECHNIQUE: ON NOETHERIAN INVOLUTION RINGS AND SYMMETRY OF PRIMITIVITY

Boris Širola

Abstract.   Let θ : R → S be a ring anti-isomorphism. We study θ-homomorphisms between left R-modules E and right S-modules M, that is, homomorphisms of the additive groups θ : E → M satisfying θ(r.x) = θ(x).θ(r) for r R and x E. We also study the class of Noetherian involution rings and the problem of symmetry of primitivity. In particular, suppose that for every semiprimitive Noetherian involution ring which has exactly two minimal prime ideals both of these satisfy (SP). Then every prime ideal of an arbitrary Noetherian ring satisfies (SP); we say that a prime ideal P of some ring satisfies (SP), the symmetry of primitivity, if it holds that P is left primitive if and only if it is right primitive. Besides, as an interesting fact, we note that any factor ring of the enveloping algebra of the Lie algebra (2) over a field of characteristic zero is an involution algebra, and so it satisfies the Krull symmetry.

2000 Mathematics Subject Classification.   16W10, 16D60.

Key words and phrases.   Antihomomorphism, antiautomorphic ring, involution ring, θ-homomorphism, prime ideal, primitive ideal, symmetry of primitivity.

DOI: 10.3336/gm.40.1.04

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