near-ring in nLab

Near-rings

Near-rings

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

Canceling xx on the left and yy on the right, we have x+y=y+xx+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 RR equipped with

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

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

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

It follows from the definition that for every x∈Rx \in R, we have 0⋅x=00 \cdot x=0 since 0⋅x=(0+0)⋅x=(0⋅x)+(0⋅x)0 \cdot x=(0+0) \cdot x=(0 \cdot x)+(0\cdot x).

If (R,+,0)(R,+,0) is only a monoid or semigroup, we say instead that RR 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)(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)(G,+,0) be a (non necessarily abelian) group. Then, the set ℱ(G)\mathcal{F}(G) of all functions from GG to GG is a near-ring with f+gf+g defined by (f+g)(x)=f(x)+g(x)(f+g)(x)=f(x)+g(x), 00 defined by 0(x)=00(x)=0, f⋅gf \cdot g defined by (f⋅g)(x)=f(g(x))(f \cdot g)(x)=f(g(x)) for every x∈Xx \in X and 1:G→G1: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