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417:, so any semimodule over a ring is in fact a module. Any semiring is a left and right semimodule over itself in the same way that a ring is a left and right module over itself. Every commutative monoid is uniquely an
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359:-semimodule can be defined similarly. For modules over a ring, the last axiom follows from the others. This is not the case with semimodules.
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consists of an additively-written commutative monoid
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344:{\displaystyle 0_{R}m=r0_{M}=0_{M}}
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476:Semirings and their Applications
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520:. You can help Knowledge by
472:"Semimodules over semirings"
454:{\displaystyle \mathbb {Z} }
432:{\displaystyle \mathbb {N} }
203:{\displaystyle (r+s)m=rm+sm}
149:{\displaystyle r(m+n)=rm+rn}
470:Golan, Jonathan S. (1999),
254:{\displaystyle (rs)m=r(sm)}
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387:is an additive inverse of
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85:{\displaystyle R\times M}
516:-related article is a
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410:{\displaystyle m\in M}
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577:Linear algebra stubs
567:Algebraic structures
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284:{\displaystyle 1m=m}
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37:commutative monoid
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51:Formally, a
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375:, then any
18:mathematics
561:Categories
491:2022-02-22
465:References
60:semimodule
47:Definition
22:semimodule
461:-module.
402:∈
77:×
391:for all
363:Examples
355:A right
26:semiring
24:over a
482:
98:axioms
512:This
371:is a
518:stub
480:ISBN
373:ring
53:left
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