Introduction

Abstract

Abstract The motivation for a study of the properties of categories of commutative algebras was the natural outcome of the publication of several papers developing techniques which mimic those of classical commutative algebral and algebraic geometry in areas such as the study of ordered algebra, lattice algebra, real algebra, differential algebra, and C∞ -algebra. For instance, G. W. Brumfiel’s Partially ordered rings and semi-algebraic geometry [4] develops a kind of ordered commutative algebra, the paper by K. Keimel entitled ‘The representation of lattice ordered groups and rings by sections in sheaves’ [28] develops a kind of lattice-ordered commutative algebra, the paper by M. E. Alonso and M. F. Roy entitled ‘Real strict localizations’ [1] develops real commutative algebra, the paper by W. K. Keigher entitled ‘Prime differential ideals in differential rings’ [27] develops differential commutative algebra and the paper by I. Moerdijk and G. E. Reyes ‘Rings of smooth functions and their localizations’ [32] develops C∞ commutative algebra. There are many other similar papers intent on extending the domain of interpretation of commutative algebra and transcending the particularity of ring structure. Thus the idea arose that the essence of commutative algebra is to be sought not in its calculus in commutative rings, but rather in the universal calculus in the category of commutative rings. It became natural to look for the universal properties in categories of commutative algebras which make commutative algebra work.