R denotes always a radical algebra over a field φ. A left ring ideal of R which is also a subvector space over φ is called a left algebra ideal of R. R is said to be left algebra noetherian if it satisfies the ascending chain condition for left algebra ideals. If dim R < ∞, then(i) R is finitely generated(ii) R is left alehra noetherian(iii) R is algebraic.Since the radical of an algebraic algebra is nil ([4] P. 19), conditions (i), (ii), (iii) are also sufficient for R to be finite-dimensional.