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.
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).
0 is a zero for ·: 0·x = 0 and x·0 = 0.
(Z, + ,−,0,·,1), the ring of integers with addition, subtraction, zero, multiplication, and one.
|Congruence n-permutable||yes, n = 2|
|Congruence extension property|
|Definable principal congruences|
|Equationally definable principal congruences|
|Strong amalgamation property|
|Epimorphisms are surjective|