Algebras of homological dimension 1

Augmented algebras over a field of homological dimension 1 ($\operatorname{hd}R=1$) are studied. It is proved that if $\operatorname{hd}R=1$, then the associated graded algebra $E(R)$ is free. If the filtration of the algebra $R$ defined by the powers of the augmentation ideal is separated, then the following conditions are equivalent: 1) $\operatorname{hd}R=1$, 2) $E(R)$ is free, 3) $\operatorname{w.g.dim}R=1$.
Some properties of groups of homological dimension 1 are presented.
It is proved that, in the category of graded algebras, the functor that produces homology groups carries a direct sum into a free product and a free product into a direct sum.
Bibliography: 6 titles.