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# Envelope Dimension of Modules and the Simplified Radical Formula

Published:2012-09-21
Printed: Dec 2013
• A. Nikseresht,
Department of Mathematics, College of Sciences, Shiraz University, 71457-44776, Shiraz, Iran
• A. Azizi,
Department of Mathematics, College of Sciences, Shiraz University, 71457-44776, Shiraz, Iran
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## Abstract

We introduce and investigate the notion of envelope dimension of commutative rings and modules over them. In particular, we show that the envelope dimension of a ring, $R$, is equal to that of the $R$-module $R^{(\mathbb{N})}$. Also we prove that the Krull dimension of a ring is no more than its envelope dimension and characterize Noetherian rings for which these two dimensions are equal. Moreover we generalize and study the concept of simplified radical formula for modules, which we defined in an earlier paper.
 Keywords: envelope dimension, simplified radical formula, prime submodule
 MSC Classifications: 13A99 - None of the above, but in this section 13C99 - None of the above, but in this section 13C13 - Other special types 13E05 - Noetherian rings and modules