# Semirings with identity and zero

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### Definition

A semiring with identity and zero is a structure S = (S, + ,0,·,1 ) of type (2,0,2,0) such that
(S, + ,0) is a commutative monoid
(S,·,1) is a monoid
0 is a zero for ·:   x = 0  and  x·0 = 0
· distributes over + :   x·(y + z) = x·y + x·z  and  (y + zx = y·x + z·x.

### Morphisms

Let S and T be semirings with identity and zero. A morphism from S to T is a function h : ST that is a homomorphism: h(x + y) = h(x) + h(y)  and  h(x·y) = h(xh(y)  and  h(0) = 0  and  h(1) = 1.

### Properties

 Classtype variety Equational theory decidable Quasiequational theory First-order theory undecidable Locally finite no Residual size unbounded Congruence distributive no Congruence modular no Congruence n-permutable Congruence regular Congruence uniform Congruence extension property Definable principal congruences Equationally definable principal congruences Amalgamation property Strong amalgamation property Epimorphisms are surjective

[Size 1]?:  1
[Size 2]?:  1
[Size 3]?:
[Size 4]?:
[Size 5]?:
[Size 6]?:

### Subclasses

Idempotent semirings with identity and zero
Rings with identity

### Superclasses

Semirings with zero
Semirings with identity

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