Note on Relative Homological Dimension

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Abstract

<jats:p>Let <jats:italic>R</jats:italic> be a ring with identity element 1, and let <jats:italic>S</jats:italic> be a subring of <jats:italic>R</jats:italic> containing 1. We consider <jats:italic>R</jats:italic>-modules on which 1 acts as the identity map, and we shall simultaneously regard such <jats:italic>R</jats:italic>-modules as <jats:italic>S</jats:italic>-modules in the natural way. In [4], we have defined the relative analogues <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0027763000023539_inline1" /> of the functors <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0027763000023539_inline2" /> of Cartan-Eilenberg [1], and we have briefly treated the corresponding relative analogues of module dimension and global ring dimension.</jats:p>

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