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Semimodule
In mathematics, a semimodule over a semiring ''R'' is an algebraic structure analogous to a module over a ring, with the exception that it forms only a commutative monoid with respect to its addition operation, as opposed to an abelian group. Definition Formally, a left ''R''-semimodule consists of an additively-written commutative monoid ''M'' and a map from R \times M to ''M'' satisfying the following axioms: # r (m + n) = rm + rn # (r + s) m = rm + sm # (rs)m = r(sm) # 1m = m # 0_R m = r 0_M = 0_M. A right ''R''-semimodule can be defined similarly. For modules over a ring, the last axiom follows from the others. This is not the case with semimodules. Examples If ''R'' is a ring, then any ''R''-module is an ''R''-semimodule. Conversely, it follows from the second, fourth, and last axioms that (−1)''m'' is an additive inverse In mathematics, the additive inverse of an element , denoted , is the element that when added to , yields the additive identity, 0 (zero). In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiring
In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distributive lattices. The smallest semiring that is not a ring is the two-element Boolean algebra, for instance with logical disjunction \lor as addition. A motivating example that is neither a ring nor a lattice is the set of natural numbers \N (including zero) under ordinary addition and multiplication. Semirings are abundant because a suitable multiplication operation arises as the function composition of endomorphisms over any commutative monoid. Terminology Some authors define semirings without the requirement for there to be a 0 or 1. This makes the analogy between ring and on the one hand and and on the other hand work more smoothly. These authors often use rig for the concept defined here. This originated as a joke, suggestin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
