# Definition:Properties of Algebraic Structures of Two Operations

## Definition

The purpose of this page is to gather into one place the various types of algebraic structure of two binary operations: $+, *$, and to classify them according to the properties they hold.

The properties are defined by the following key:

 $A0$ Closure of $+$ $A1$ Associativity of $+$ $A2$ Identity Element of $+$ $A3$ Inverses of $+$ $A4$ Commutativity of $+$ $M0$ Closure of $+$ $M1$ Associativity of $*$ $M2$ Identity Element of $*$ $M3$ Inverses of $*$ $M4$ Commutativity of $*$ $D$ Distributivity of $*$ over $+$ $NZD$ No (Proper) Zero Divisors $AC2$ Characteristic 2 for $+$ $AZ$ Zero Element of $+$ $D'$ Distributivity of $+$ over $*$ $MI$ Idempotence of $*$ $MZ$ Zero Element of $*$

$A0$ $A1$ $A2$ $A3$ $A4$ $M0$ $M1$ $M2$ $M3$ $M4$ $D$ $NZD$ $AC2$ $AZ$ $MI$ $MZ$ $D'$
Semiring $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
Additive Semiring $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
Rig $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
Ring without associativity $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $(\checkmark)$ (see here)
Ring $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $(\checkmark)$ (see here)
Ring with Unity $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $(\checkmark)$ (see here)
Commutative Ring with Unity $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $(\checkmark)$ (see here)
Integral Domain $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $(\checkmark)$ (see here)
Division Ring $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $(\checkmark)$ (see here)
Field $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $(\checkmark)$ (see here)
Idempotent Ring $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
Boolean Ring $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
Boolean Algebra $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$

A checkmark in brackets: $(\checkmark)$ denotes that the property indicated can be derived from the others.