Commutative ideal theory without finiteness conditions: Completely irreducible ideals

Fuchs, Laszlo; Heinzer, William; Olberding, Bruce

doi:10.1090/S0002-9947-06-03815-3

Commutative ideal theory without finiteness conditions: Completely irreducible ideals
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by Laszlo Fuchs, William Heinzer and Bruce Olberding PDF

Trans. Amer. Math. Soc. 358 (2006), 3113-3131 Request permission

Abstract:

An ideal of a ring is completely irreducible if it is not the intersection of any set of proper overideals. We investigate the structure of completely irrreducible ideals in a commutative ring without finiteness conditions. It is known that every ideal of a ring is an intersection of completely irreducible ideals. We characterize in several ways those ideals that admit a representation as an irredundant intersection of completely irreducible ideals, and we study the question of uniqueness of such representations. We characterize those commutative rings in which every ideal is an irredundant intersection of completely irreducible ideals.

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