On left ideal essential extensions of rings
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ON LEFT IDEAL ESSENTIAL EXTENSIONS OF RINGS M. NOWAKOWSKA1,∗ and E. R. PUCZY�LOWSKI2 1
Institute of Mathematics, University of Silesia in Katowice, Katowice, Poland e-mail: [email protected] 2
Institute of Mathematics, University of Warsaw, Warsaw, Poland e-mail: [email protected] (Received December 12, 2019; accepted April 10, 2020)
Abstract. The main goal of this paper is to extend Flanigan’s theorem in [4] concerning ideal essential extensions of rings to left ideal essential extensions. Moreover, we give new proofs of two Flanigan’s theorems and answer a question raised by M. Petrich in [5] which is related to pure extensions of rings.
1. Introduction All rings in this paper are associative but not necessarily with unity. For a given ring R and a subset X of R, we denote by lR (X) = {r ∈ R | rX = 0} the left annihilator of X, by rR (X) = {r ∈ R | Xr = 0} the right annihilator of X and by aR (X) = {r ∈ R | rX = Xr = 0} the annihilator of X. We write A ⊳ R (A
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