Equations over universal algebras: some old and new surprises

Abstract:
Solving equations is in the center of algebra and has induced several new number systems and algebraic structures. In the variety of commutative rings with identity, any solution of a polynomial equation p = 0 is also a zero of any polynomial in the ideal (p) generated by p; hence we might speak of "the zero of an ideal". Let us call an equation (or a system of equations) over a ring R "sovable" if there exists an extension ring S which contains a solution of this equation (of these equations). Hence x^2+1=0 is solvable over the reals. A version of the Nullstellensatz answers the question when a polynomial equation p=0 over R is solvable in this sense: this is the case iff (p) has zero intersection with R. So p = 2x+1 = 0 is not solvable over the integers modulo 4 since p^2 = 1. If we leave this well-known (?) variety, we face lots of surprises. For instance, x^2+1=0 has uncountably (!) many solution in the Hamiltonians. If A ž B, an equation which is solvable over A need not (!) be solvable over B, and every system of equations in a variety has either no solution, or precisely one, or the number of solutions exceeds every given cardinal in a suitable extension - a dramatic contrast to the case of fields.