  Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a
standard graded $S$-algebra. In terms of the Gr\"obner basis of the defining
ideal $J$ of $A$ we give a condition, called the x-condition, which implies
that all graded components $A_k$ of $A$ have linear quotients and with
additional assumptions are componentwise linear. A typical example of such an
algebra is the Rees ring $R(I)$ of a graded ideal or the symmetric algebra
$Sym(M)$ of a module $M$. We apply our criterion to study certain symmetric
algebras and the powers of vertex cover ideals of certain classes of graphs.

Commutative Algebra