It is proved that two diagonal matrices diag(a 1 , . . . , a n ) and diag(b 1 , . . . , b n ) over a local ring R are equivalent if and only if there are two permutations σ, τ of {1, 2, . . . , n} such that [R/a i R] l = [R/b σ (i) R] l and [R/a i R] e = [R/b τ (i) R] e for every i = 1, 2, . . . , n. Here [R/aR] e denotes the epigeny class of R/aR, and [R/aR] l denotes the lower part of R/aR. In some particular cases, like for instance in the case of R local commutative, diag(a 1 , . . . , a n ) is equivalent to diag(b 1 , . . . , b n ) if and only if there is a permutation σ of {1, 2, . . . , n} with a i R = b σ (i) R for every i = 1, . . . , n. These results are obtained studying the directsum decompositions of finite direct sums of cyclically presented modules over local rings. The theory of these decompositions turns out to be incredibly similar to the theory of direct-sum decompositions of finite direct sums of uniserial modules over arbitrary rings.

An R-module M is called coretractable if Hom R (M/K , M) = 0 for any proper submodule K of M. In this paper we study coretractable modules and their endomorphism rings. It turns out that if all right R-modules are coretractable, then R is a right Kasch and (two-sided) perfect ring. However, the converse holds for commutative rings. Also, for a semi-injective coretractable module M R with S = End R (M), we show that u.dim( S S) = corank(M R ).2000 Mathematics subject classification: primary 16D10; secondary 16D40, 16L30.

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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).

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