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2012•arXiv (Cornell University)Open access

On Left FGC-Rings and a Partial Solution of Kaplansky's Problem on Duo-Rings ∗†‡

Gholamreza Behboodi Eskandari, Mahmood Behboodi

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Abstract

In this paper we study (non-commutative) rings R over which every finitely generated left module is a direct sum of cyclic modules (called left FGC-rings). The commutative case was a well-lnown problem studied and solved in 1970s by various authors. The main result of this paper shows that a Noetherian local left FGC-ring is either an Artinian principal left ideal ring, or an Artinian principal right ideal ring, or a prime ring over which every two-sided ideal is principal as a left and a right ideal. As a consequence, we obtain that if R = � n=1 Ri is a finite product of Noetherian duo-rings Ri where each Ri is prime or local, then R is a left FGC-ring if and only if R is a principal ideal ring.

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What this paper is about

In this paper we study (non-commutative) rings R over which every finitely generated left module is a direct sum of cyclic modules (called left FGC-rings). The commutative case was a well-lnown problem studied and solved in 1970s by various authors. The main result of this paper shows that a Noetherian local left FGC-ring is either an Artinian principal left ideal ring, or an Artinian principal right ideal ring, or a prime ring over which every two-sided ideal is principal as a left and a right ideal. As a consequence, we obtain that if R = � n=1 Ri is a finite product of Noetherian duo-rings Ri where each Ri is prime or local, then R is a left FGC-ring if and only if R is a principal ideal ring.

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Available abstract

In this paper we study (non-commutative) rings R over which every finitely generated left module is a direct sum of cyclic modules (called left FGC-rings). The commutative case was a well-lnown problem studied and solved in 1970s by various authors. The main result of this paper shows that a Noetherian local left FGC-ring is either an Artinian principal left ideal ring, or an Artinian principal right ideal ring, or a prime ring over which every two-sided ideal is principal as a left and a right ideal. As a consequence, we obtain that if R = � n=1 Ri is a finite product of Noetherian duo-rings Ri where each Ri is prime or local, then R is a left FGC-ring if and only if R is a principal ideal ring.

Key concepts: Mathematics, Principal ideal ring, Noetherian ring, Ideal (ethics), Radical of a ring, Primary ideal, Noncommutative ring, Prime (order theory)

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On Left FGC-Rings and a Partial Solution of Kaplansky's Problem on Duo-Rings ∗†‡ — Research Paper | ScholarLens