Abstract
A subgroup I of (R, +) is called left (right) ideal if
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R·I = {ri|r ∈ R, i ∈ I} ⊆ I (resp. I·R ⊆ I) and ideal if it is left and right ideal. We denote by (X) = ∩ {I (left or right) ideal in R|X ⊆ I}, if X ⊆ R, called the (left or right) ideal generated by X. In an arbitrary ring \( \left( X \right) = \left\{ {\sum\limits_{i = 1}^n {{r_i}{x_i} + \sum\limits_{k = 1}^m {{{x'}_k}{{r'}_k}} } + \sum\limits_{s = 1}^l {{{r''}_s}{{x''}_s}} {{r'''}_s} + \sum\limits_{j = 1}^t {{n_j}{y_j}}}\right\}\)
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with \( {r_i},{r'_{k,}}{r''_s},{r'''_s} \in R,{x_i},{x'_k},{x''_s},{y_j} \in X,{n_j} \in \) and the reader can provide the simplier forms if the ring has identity or is commutative (or both).
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© 1998 Springer Science+Business Media Dordrecht
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Cǎlugǎreanu, G., Hamburg, P. (1998). Ideals. In: Exercises in Basic Ring Theory. Kluwer Texts in the Mathematical Sciences, vol 20. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9004-4_2
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DOI: https://doi.org/10.1007/978-94-015-9004-4_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4985-8
Online ISBN: 978-94-015-9004-4
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