On essential extensions of direct sums of injective modules

Abstract.

We characterize locally Noetherian modules M _R in terms of essential extensions of direct sums of M-injective modules. As a special case (M = R) we obtain that the following conditions are equivalent: (a) R is right Noetherian; (b) any essential extension of the direct sum of any family of injective right R-modules is the direct sum of injective right R-modules; (c) any essential extension of the direct sum of the family $\{{\cal E}(S_i) \, | \, i = 1, 2, ...\}$ of injective hulls of any family $\{S_i \, | \, i = 1, 2, ...\}$ of simple right R-modules is the direct sum of injective right R-modules; (d) for any family¶$\{S_i \, | \, i = 1, 2, ...\}$ of simple right R-modules there exists an infinite subset ${\cal I}$ of natural numbers such that $\oplus_{i \in{\cal I}}{\cal E}(S_i)$ is an injective module.