Some extensions of the theory of semirings
Dean, Robert Gross
Dean, Robert Gross
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1966
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Abstract
A set S on which are defined two binary operations, called addition and multiplication, is a semiring if S is a semigroup under each operation and if multiplication distributes over addition from both the left and the right, In this paper, some properties which relate semirings to rings and some properties which distinguish semirings from rings are studied. Extensions are made to the general theory of semirings. Among these extensions are some dealing with the properties of an additive identity, with the effect of positive characteristic, with additive regularity, and with the persistence of these properties in direct sums, where the summands have such properties. A generalization of certain aspects of the theory of p-rings to p-semirings and n-semirings is made. An element is called purely n-potent if it is n-potent and is not q-potent for 2 <= q < n. Three types of semirings are defined as generalizations of the notion of a p-ring. The first, in which each element is p-potent and which has characteristic p, is called a char p-semiring. The second, in which each non-identity is purely n-potent, is called a purely n-semiring. The third, in which each element is n-potent and in which there exists at least one purely n-potent element, is called a weak n-semiring. A new characterization for Boolean rings and p-rings is given. A mutually distributive semiring, in which each operation distributes over the other, is defined and investigated. The relation between mutually distributive semirings and Boolean semirings is determined. A proof is given that there is no mutually distributive field. A new mapping of a semiring, a skewmorphism, which reverses operations, is defined. Sufficient conditions are determined for a semiring to be mutually distributive. A structure analogous to the ideal, the cell, is introduced. Cosets modulo a cell are shown to form a semiring in which the cell is an all element. The problem of a homomorphism theorem analogous to the fundamental theorem of homomorphism for semirings proved by S. Bourne is studied. In a semiring S, if phi is a homomorphism of S onto a semiring S' with all element, the set of elements C which are mapped by phi onto the all element of S' form a cell in s. The quotient semiring S/C is unity-semi-isomorphic to S'.
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Rings (Algebra)
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Dissertation
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iv, 54 leaves, bound : illustrations
Department
Mathematics