Measures and measurable functions are used primarily as tools for carrying out various calculations to increase our knowledge. We learn how to combine them in various ways by studying real analysis; a very useful subject on which very much has been written. In this paper, we regard measurable functions as algebras of real-valued functions (or equivalence classes of them) on a set or topological space under point-wise addition, multiplication, or lattice operations and our techniques resemble closely those used to study algebras of continuous functions. This is done by examining a number of explicit examples including Borel and Lebesgue measures and measurable functions.
We associate a graph Γ+(R) to a ring R whose vertices are nonzero proper right ideals of R and two vertices I and J are adjacent if I + J = R. Then we try to translate properties of this graph into algebraic properties of R and vice versa. For example, we characterize rings R for which Γ+(R) respectively is connected, complete, planar, complemented or a forest. Also we find the dominating number of Γ+(R).
Abstract. Let R be a commutative ring with identity and M an Rmodule. In this paper, we associate a graph to M , say Γ(M ), such that when M = R, Γ(M ) is exactly the classic zero-divisor graph.
In this article, we study essential ideals, socle, and some related ideals of rings of real valued measurable functions. We also study the Goldie dimension of rings of measurable functions.
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