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Published in:
2003 | OriginalPaper | Chapter
Symmetric extension of a semiring
Author : Jonathan S. Golan
Published in: Semirings and Affine Equations over Them: Theory and Applications
Publisher: Springer Netherlands
Included in: Professional Book Archive
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Let R be a semiring and let I ∈ ideal(R). Then it is easy to verify that $${{\mathcal{D}}_{2}}\left( {R,I} \right) = \left\{ {\left[ {\begin{array}{*{20}{c}} a & b \\ b & a \\ \end{array} } \right] \in {{\mathcal{M}}_{2}}\left( R \right)\left| {a \in R\,and\,b \in I} \right.} \right\}$$ is a subsemiring of M2(R). This semiring is additively idempotent if and only if R is, and is commutative if and only if R is.