it is shown that a commutative reduced ring r is a baer ring if and only if it is a cs-ring; if and only if every dense subset of spec (r) containing max (r) is an extremally disconnected space; if and only if every non-zero ideal of r is essential in a principal ideal generated by an idempotent.

In this paper we give conditions for a tuple of commutative bounded linear operators which holds in the property of the Hypercyclicity Criterion. We characterize topological transitivity and semi-hereiditarily of a dynamical system given  by an n-tuple of operators acting on a separable infinite dimensional Banach space .

Totalistic cellular automata were introduced by Wolfram [1, 2] as a means of studying higher-color cellular automata within a reasonable space of possible rules. The underlying principle is to systematically construct a subset of rules within all the possible rules of the highercolor cellular automata, and the system in totalistic cellular automata is to add up the values of the cells within th...