Wang, Huaxiong Injective hulls of semimodules over additively-idempotent semirings. (English) Zbl 0802.16041 Semigroup Forum 48, No. 3, 377-379 (1994). A semiring \((R,+,\cdot)\) is a nonempty set \(R\) where \((R,+)\) is a commutative monoid with additive identity 0, \((R,\cdot)\) is a monoid with multiplicative identity 1, \(0r = 0 = r0\) for all \(r\in R\) and multiplication distributes over addition from either side. A semiring \(R\) is said to be additively-idempotent, if \(r + r = r\) for all \(r \in R\).A commutative monoid \((M,+,0_ M)\) is called a left semimodule over a semiring \(R\) (briefly left \(R\)-semimodule) if there is a scalar multiplication \((r,m) \to rm\) from \(R \times M\) to \(M\) satisfying standard conditions. A left \(R\)-semimodule \(I\) is said to be injective if, given a subsemimodule \(N\) of a semimodule \(M\), any \(R\)-homomorphism from \(N\) to \(I\) can be extended to an \(R\)-homomorphism from \(M\) to \(I\).It is well known that every left module over a ring \(R\) can be embedded into an injective left \(R\)-module. In the present paper the analogous result is obtained for a left semimodule over an additively-idempotent semiring. Reviewer: B.Pondělíček (Praha) Cited in 8 Documents MSC: 16Y60 Semirings 16D50 Injective modules, self-injective associative rings Keywords:injective left semimodules; left semimodule; additively-idempotent semiring PDFBibTeX XMLCite \textit{H. Wang}, Semigroup Forum 48, No. 3, 377--379 (1994; Zbl 0802.16041) Full Text: DOI EuDML References: [1] Brungs, G., and H. Lakser,Injective hulls of semilattices, Canad. Math. Bull.13 (1975), 297–298. [2] Cohn, P. M., ”Universal Algebra,” Harper & Row, New York, 1965. · Zbl 0141.01002 [3] Golan, J. S., ”The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science”, Longman, Harlow, 1992. · Zbl 0780.16036 [4] Joyal, A., and M. Tierney, ”An Extension of the Galois Theory of Grothendieck”, A. M. S. Memoir #309, Amer. Math. Soc., Providence, 1984. · Zbl 0541.18002 [5] Takahashi, M., and H. Wang,Injective semimodule over a 2-semiring, Kobe J. Math.10 (1993), 59–70. · Zbl 0814.16043 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.