The present work is concerned with what are called polynomial algebras as an extension of the work of Ramakrishnan and his colleagues on the algebras of matrices satisfying conditions like Lm = I and Lm = Lk. Assuming Lm to be an m-dimensional linear space, we generate a class of associative algebras called polynomial algebras by requiring that every element L of Lm satisfy a polynomial equation Ln + P1Ln−1 + ⋯ + Pn = 0. We show that some very important algebras which physicists have found useful can be obtained by various restrictions on the polynomial. A few general properties of these algebras are established.