What is the largest variety of ring-like algebras in which the additive identity is absorbing, without it being an axiom?

Question

When I say "ring-like algebra" I mean two binary operations, "addition" and "multiplication", such that multiplication distributes over addition on at least one side, and addition has at least a one-sided additive identity, but no other requirements are imposed - i.e. neither operation need be associative, commutative, etc.

My goal is to find the minimal axioms not themselves including that the additive identity is absorbing for multiplication, which are sufficient to prove that it is. By minimal, I mean proving the "fewest" theorems - that is, the resulting set of theorems being a proper subset of any set of theorems describing any other variety of algebra which have this property. A better way to say it may be I'm looking for the largest variety in which every algebra satisfies these constraints. (I'm still getting used to the terminology.)

So far I have the following:

Given the signature (S,+,*,(-),0), where S is a set, + and * are binary functions S×S→S (with * written by concatenation), (-) is a unary function S→S, and 0 is a constant element of S, and the following axioms (all variables universally quantified over elements of S):

1. a = 0 + a
2. (a+b)c = ac + bc
3. b = (b+a) + (-a)

then we can prove:

1. a = 0 + a
2. ab = (0+a)b = 0b + ab
3. ab + (-ab) = (0b+ab) + (-ab)
4. ab + (-ab) = 0b
5. (0+ab) + (-ab) = 0b
6. 0 = 0b

What I want to know is, is there a larger variety containing the first two axioms in which the same result can be proven? More generally, can some variety be proven to be the largest for which every algebra satisfies these requirements?

Answer 1

Your axioms are identities. A class defined by identities is a variety. A stronger set of identities defines a smaller variety, and a weaker set of identities defines a larger variety.

You have asked whether there is a largest variety satisfying the identities

(1) $a=0+a$, and
(2) $(a+b)c = ac+bc$,

in which the following identity is true:

(4) $0=0b$.

There is a largest such variety, namely the variety axiomatized by identities (1), (2), and (4).

You have also asked whether this largest variety has a set of axioms involving (1), (2) and some other axioms, but not involving (4). There is such a set of axioms, namely the set containing (1), (2), and (5), where we define (5) to be

(5) $0=0(0+b)$.

It is easy to see that the varieties defined by (1), (2), (4) is the same as the one defined by (1), (2), and (5).

The set of axioms consisting of (1), (2) and (5) does not imply your axiom

(3) $b = (b+a) + (-a)$

since (1), (2), and (5) say nothing about the behavior of the negation symbol, so it is easy to construct structures satisfying (1), (2) and (5) which have a strange negation that does not satisfy (3). Thus (1), (2) and (5) are a strictly weaker set of axioms than (1), (2), and (3).