nLab possibly empty ring

Idea

By replacing every element in the definition with a constant function to the element, the empty set vacuously satisfies the axioms of any algebraic structure.

Definition

A possibly empty ring is a set $R$ with a binary operation $(-)+(-):R \times R \to R$, a unary operation $-:R \to R$, a unary operation $0:R \to R$, a binary operation $(-)\cdot(-):R \times R \to R$, and a unary operation $1:R \to R$ such that that:

• For all $a$, $b$, and $c$ in $R$, $a+(b+c)=(a+b)+c$
• For all $a$ and $b$ in $R$, $a+b=b+a$
• For all $a$ and $b$ in $R$, $a+0(b)=a$
• For all $a$ and $b$ in $R$, $a+(-a)=0(b)$
• For all $a$, $b$, and $c$ in $R$, $a\cdot (b\cdot c)=(a\cdot b)\cdot c$
• For all $a$ and $b$ in $R$, $a\cdot 1(b)=a$
• For all $a$ and $b$ in $R$, $1(a)\cdot b=b$
• For all $a$, $b$, and $c$ in $R$, $a\cdot (b+c)= a\cdot b + a\cdot c$
• For all $a$, $b$, and $c$ in $R$, $(a+b)\cdot c= a\cdot c + b\cdot c$

A possibly empty ring is commutative if for all $a$ and $b$ in $R$, $a\cdot b=b \cdot a$.

Examples

• Every ring is a possibly empty ring.

• The empty possibly empty ring is an possibly empty ring that is not a ring.

Related concepts

• ring

• possibly empty field?