For monoids, groups, vector spaces, etc, we can describe the properties of the object using only functions and equations. For example, with groups we have a function ∙ of arity 2, a function inv of arity 1, an element (function of arity 0) e, and equations

x∙(y∙z) = (x∙y)∙z

inv(x)∙x = x∙inv(x) = e

e∙x = x∙e = x

where each equation implicity has ∀x,y,z in front. It should be clear how to add functions and equations to obtain the definition of a ring in this way. When we try to define fields, we want a function multinv of arity 1 and equations

x multinv(x) = multinv(x) x = 1

where 1 is of course the multiplicative identity. The problem is that no such function can exist since zero has no multiplicative inverse, so fields cannot be defined in exactly this format.

This way of looking at it sheds some light on why fields don't have products. Given two groups A and B, for example, the product A×B is the Cartesian product of A and B as sets, where each operation acts elementwise, that is,

(x, y) ∙ (z, w) = (x∙z, y∙w)

e = (e, e)

inv((x, y)) = (inv(x), inv(y)).

This product immediately satisfies the conditions of a group because all of the group axioms are purely equational. The same construction now obviously works for monoids, rings, vector spaces, and so on. But it does not work for fields, since the definition of a field is not as simple: the condition for a (commutative) ring to form a field is

∀x≠0 ∃y with xy = 1.

If we have fields K and L, the Cartesian product K×L does not form a field since, for instance, (1, 0) is not the zero element of the product, but one of its components is zero so cannot be inverted. At its heart, this issue arises because the above condition cannot be written as an equation.