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The paper studies the regions of values of the systems {f(z1), f(r1), f(r2),..., f(rn)} and {f(r1), f(r2),..., f (rn)}, where n ⁥ 2; z1 is an arbitrary fixed point of the disk U = {z: |z| < 1} with Im z1 ≠ 0; rj are fixed numbers, 0 < rj < 1, j = 1, 2,..., n; f ∈ T, and the class T consists of the functions f(z), f(0) = 0, f′(0) = 1, regular in the disk U and satisfying the condition Im f(z) · Imz > 0 for Im z ≠ 0. As an implication, the region of values of f(z1) in the subclass of functions f ∈ T with prescribed values f(rj) (j = 1, 2,..., n) is determined. Bibliography: 12 titles. [ABSTRACT FROM AUTHOR]

A right R–module E over a ring R is said to be maximally injective in case for any maximal right ideal m of R, every R–homomorphism f : m → E can be extended to an R–homomorphism f' : R → E. In this paper, we first construct an example to show that maximal injectivity is a proper generalization of injectivity. Then we prove that any right R–module over a left perfect ring R is maximally injective if and only if it is injective. We also give a partial affirmative answer to Faith's conjecture by further investigating the property of maximally injective rings. Finally, we get an approximation to Faith's conjecture, which asserts that every injective right R–module over any left perfect right self–injective ring R is the injective hull of a projective submodule. [ABSTRACT FROM AUTHOR]