An axiomatic approach to block decompositions of rings

In calculations with matrices, block calculations play an important role. To elucidate the essential structures of block decompositions, we shall in this paper introduce a simple system of axioms which guarantees block calculations of rings. The axioms can be interpreted as rules of some kind of information filters. We shall also give another system of axioms in terms of idempotents which is equivalent to the above mentioned system and is similar to the Kolmogorov axioms of probability spaces.