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The idempotent divisor graph of a commutative ring R is a graph with a vertex set in R* = R-{0}, where any distinct vertices x and y are adjacent if and only if x.y = e, for some non-unit idempotent element e^2 = e ∈ R, and is denoted by Л(R). Our goal in this work is to transform the planar idempotent divisor graph after coloring its regions into optical art by depending on the reflection of vertices, edges, and planes on the x or y-axes. That is, we achieve Op art solely through pure mathematics in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we study on modules that have a weak (ample) δ-supplement in every extension which are adapted Zöschinger's modules with the properties (E) and (EE). It is shown that: (1) Direct summands of modules with the property δ-(CWE) have the property δ-(CWE); (2) For a module M, if every submodule of M has the property δ-(CWE) then so does M; (3) For a ring R, R is δ-semilocal iff every R-module has the property δ-(CWE); (4) Every factor module of a finitely generated module that has the property δ-(CWE) also has the property δ-(CWE) under a special condition; (5) Let M be a module and L be a submodule of M such that L <<#948; M. If the factor module M/L has the property δ-(CWE), then so does M; (6) On a semisimple module the concepts of modules that have the property δ-(CE) and δ-(CWE) coincide with each other. [ABSTRACT FROM AUTHOR]