Fibred product of commutative algebras: generators and relations

The method of direct computation of universal (fibred) product in the category of commutative associative algebras of finite type with unity over a field is given and proven. The field of coefficients is not supposed to be algebraically closed and can be of any characteristic. Formation of fibred product of commutative associative algebras is an algebraic counterpart of gluing algebraic schemes by means of some equivalence relation in algebraic geometry. If initial algebras are finite-dimensional vector spaces the dimension of their product obeys Grassmann-like formula. Finite-dimensional case means geometrically the strict version of adding two collections of points containing some common part. The method involves description of algebras by generators and relations on input and returns similar description of the product algebra. It is "ready-to-eat" even for computer realization. The product algebra is well-defined: taking another descriptions of the same algebras leads to isomorphic product algebra. Also it is proven that the product algebra enjoys universal property, i.e. it is indeed fibred product. The input data is a triple of algebras and a pair of homomorphisms $A_1\stackrel{f_1}{\to}A_0 \stackrel{f_2}{\leftarrow}A_2$. Algebras and homomorphisms can be described in any fashion. We prove that for computing the fibred product it is enough to restrict to the case when $f_i,i=1,2,$ are surjective and describe how to reduce to surjective case. Also the way to choose generators and relations for input algebras is considered.