nLab near-ring

Idea

A near-ring is a sort of ring in which addition need not be commutative. However, if we simply remove the commutativity of addition from the usual definition of a ring, then nothing changes, because we can prove commutativity from the other axioms: we have

$$(x+y)(1+1) = x(1+1) + y(1+1) = x + x + y + y$$

and also

$$(x+y)(1+1) = (x+y)1 + (x+y)1 = x + y + x + y.$$

Canceling $x$ on the left and $y$ on the right, we have $x+y=y+x$.

Thus, in order for the notion of near-ring to be different from that of a ring, we need to relax the distributivity law as well; we impose it only on one side.

Definition

A near-ring is a set $R$ equipped with

1. A group structure $(R,+,0)$,

2. A monoid structure $(R,\cdot,1)$,

3. such that for any $x,y,z\in R$ we have $(x+y)\cdot z = (x\cdot z) + (y\cdot z)$.

It follows from the definition that for every $x \in R$, we have $0 \cdot x=0$ since $0 \cdot x=(0+0) \cdot x=(0 \cdot x)+(0\cdot x)$.

If $(R,+,0)$ is only a monoid or semigroup, we say instead that $R$ is a near-rig or a near-semiring. (Of course, now it is possible to have distributivity on both sides without making addition commutative, since addition need not always cancel.)

There is also a weaker notion, of loop near ring, where in the axioms above the group structure on $(R,+,0)$ is weakened to a loop algebraic structure (i.e. left and right additive inverses no longer coincide and the associativity of addition is dropped).

Example

Let $(G,+,0)$ be a (non necessarily abelian) group. Then, the set $\mathcal{F}(G)$ of all functions from $G$ to $G$ is a near-ring with $f+g$ defined by $(f+g)(x)=f(x)+g(x)$, $0$ defined by $0(x)=0$, $f \cdot g$ defined by $(f \cdot g)(x)=f(g(x))$ for every $x \in X$ and $1:G \rightarrow G$ the identity function.

Internalization

Of course, near-rings can be defined internally to any cartesian monoidal category. More generally, they can be defined internally to a braided duoidal category.

References

• wikipedia

For loop near rings see

• D. Ramakotaiah, C. Santhakumari. On loop near-rings. Bull. Austral. Math. Soc. 19(3):417{435, 1978.

Loop near rings appear in topology:

• Damir Franetič, Petar Pavešić, Loop near-rings and unique decompositions of H-spaces, arxiv/1511.06168

category: algebra