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Interval Semirings
This book has seven chapters. In chapter one we give the basics needed to
make this book a self contained one. Chapter two introduces the notion of
interval semigroups and interval semifields and are algebraically analysed.
Chapter three introduces special types of interval semirings like matrix
interval semirings and interval polynomial semirings. Chapter four for the
first time introduces the notion of group interval semirings, semigroup
interval semirings, loop interval semirings and groupoid interval semirings and
these structures are studied. Interval neutrosophic semirings are introduced in
chapter five. Applications of these structures are given in chapter six. The
final chapter suggests around 120 problems for the reader.Comment: 155 pages; Published by Kappa & Omega in 201
Exact rings and semirings
We introduce and study an abstract class of semirings, which we call exact
semirings, defined by a Hahn-Banach-type separation property on modules. Our
motivation comes from the tropical semiring, and in particular a desire to
understand the often surprising extent to which it behaves like a field. The
definition of exactness abstracts an elementary property of fields and the
tropical semiring, which we believe is fundamental to explaining this
similarity. The class of exact semirings turns out to include many other
important examples of both rings (proper quotients of principal ideal domains,
matrix rings and finite group rings over these and over fields), and semirings
(the Boolean semiring, generalisations of the tropical semiring, matrix
semirings and group semirings over these).Comment: 17 pages; fixed typos, clarified a few points, changed notation in
Example 6.
Invertible Ideals and Gaussian Semirings
In the first section of this paper, we introduce the notions of fractional
and invertible ideals of semirings and characterize invertible ideals of a
semidomain. In section two, we define Pr\"{u}fer semirings and characterize
them in terms of valuation semirings. In this section, we also characterize
Pr\"{u}fer semirings in terms of some identities over its ideals such as for all ideals , of . In the third section, we
give a semiring version for the Gilmer-Tsang Theorem, which states that for a
suitable family of semirings, the concepts of Pr\"{u}fer and Gaussian semirings
are equivalent. At last we end this paper by giving a plenty of examples of
proper Gaussian and Pr\"{u}fer semirings.Comment: Final versio
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