Glasnik Matematicki, Vol. 45, No.2 (2010), 395-406.

VERY STRONG MULTIPLICATION IDEALS AND THE IDEAL θ (I) OVER A COMMUTATIVE SEMIRING

Shahabaddin Ebrahimi Atani and Reza Ebrahimi Atani

Department of Electrical Engineering, University of Guilan, P.O.Box 3756, Rasht, Iran
e-mail: rebrahimi@guilan.ac.ir

Abstract.   Let R a commutative semiring with identity. An ideal I is called a multiplication ideal if every ideal contained in I is a multiple of I. We consider the associated ideal θ (I). It is proved that the strong ideal θ (I) is important in the study of multiplication ideals. Among various applications given, the following results are proved: if I is a faithful very strong multiplication ideal, then the strong ideal θ (I) is an idempotent ideal of R such that θ (θ (I)) = θ (I), and every secondary representable ideal of R which is also a very strong multiplication ideal is finitely generated.

2000 Mathematics Subject Classification.   16Y60.

Key words and phrases.   Very strong multiplication ideals, ideal θ (I), secondary ideals.

DOI: 10.3336/gm.45.2.07

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