The essential point in the definition of an algebra is that it is a vector space of finite dimension over a field. This fact allows us to conclude that ascending and descending chains of subalgebras will terminate. After the great success that Emmy Noether had in her ideal theory in rings with ascending chain condition, it seemed reasonable to expect that in rings where the ascending and the descending chain condition holds for left ideals one should obtain results similar to those of Wedderburn. As one of the papers written from this point of view we mention E. Artin, Zur Theorie der hyperkomplexen Zahlen (Abh. Math. Sem. Hamburgischen Univ. vol. 5 (1926)). In 1939 C. Hopkins showed (Rings with minimal condition f or left ideals, Ann. of Math. vol. 40) that the descending chain condition suffices.

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This is a placeholder explanation generated by the low-overhead model. It interprets "The essential point in the definition of an algebra is that it is a vector space of finite dimension over a field. This fact allows us to conclude that ascending and descending chains of subalgebras will terminate. After the great success that Emmy Noether had in her ideal theory in rings with ascending chain condition, it seemed reasonable to expect that in rings where the ascending and the descending chain condition holds for left ideals one should obtain results similar to those of Wedderburn. As one of the papers written from this point of view we mention E. Artin, Zur Theorie der hyperkomplexen Zahlen (Abh. Math. Sem. Hamburgischen Univ. vol. 5 (1926)). In 1939 C. Hopkins showed (Rings with minimal condition f or left ideals, Ann. of Math. vol. 40) that the descending chain condition suffices." in the context of "(1950), "The influence of J. H. M. Wedderburn on the development of modern algebra", Bulletin of the American Mathematical Society 56 no: 1, pages: 65–72 (quote from p. 67)" to mean that wisdom is timeless.