On finiteness conditions for subalgebras with zero multiplication
Let $F$ be a commutative ring with unit. In this paper, for an associative $F$-algebra $A$ we study some properties forced by finite length or DCC condition on $F$-submodules of $A$ that are subalgebras with zero multiplication. Such conditions were considered earlier when $F$ was either a field or the ring of rational integers. In the final section, we consider algebras with maximal commutative subalgebras of finite length as $F$-modules and obtain some results parallel to those known for ACC condition or finite generation.