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#### Abbreviation: Rng_{1}

### Definition

A *ring with identity* is a structure *R* = (*R*, + ,−,0,·,1
) of type (2,1,0,2,0) such that

(*R*, + ,−,0,·) is a ring and

1 is an identity for ·: *x*·1 = *x* and 1·*x* = *x*.

### Morphisms

Let *R* and *S* be rings with identity. A morphism from *R*
to *S* is a function *h* : *R*→*S* that is a homomorphism:
*h*(*x* + *y*) = *h*(*x*) + *h*(*y*) and *h*(*x*·*y*) = *h*(*x*)·*h*(*y*) and *h*(1) = 1.

**Remark**:
It follows that *h*(0) = 0 and *h*(−*x*) = −*h*(*x*).

### Some results

0 is a zero for ·: 0·*x* = 0 and *x*·0 = 0.

### Examples

(**Z**, + ,−,0,·,1), the ring of integers with addition, subtraction, zero, multiplication, and one.

### Properties

### Finite members

[Size 1]?: 1

[Size 2]?: 1

[Size 3]?: 1

[Size 4]?: 4

[Size 5]?: 1

[Size 6]?: 1

[Finite rings with identity in the Encyclopedia of Integer Sequences]

### Subclasses

Commutative rings with identity

Division rings

### Superclasses

Rings