### Shahabaddin Ebrahimi Atani and Reza Ebrahimi Atani

Department of Mathematics, University of Guilan, P.O. Box 1914, Rasht, Iran
e-mail: ebrahimi@guilan.ac.ir

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.

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DOI: 10.3336/gm.45.2.07

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