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.

Also see

• Definition:Properties of Algebraic Structures of One Operation