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| | + | {{TEX|done}}{{MSC|16D20}} |
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| | ''double module'' | | ''double module'' |
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| − | An Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b0163101.png" /> that is a left module over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b0163102.png" /> and a right module over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b0163103.png" />, and is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b0163104.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b0163105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b0163106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b0163107.png" />. One says that this is the situation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b0163108.png" />, or that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b0163109.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631010.png" />-bimodule. The bimodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631011.png" /> may be regarded as a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631012.png" />-module, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631013.png" /> is the [[opposite ring]] (dually isomorphic, anti-isomorphic) to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631014.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631015.png" /> denotes the tensor product over the ring of integers, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631016.png" />. For every left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631017.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631018.png" /> one has the situation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631020.png" /> is the ring of endomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631021.png" />. Any ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631022.png" /> can be given the natural structure of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631023.png" />-bimodule. | + | An Abelian group $B$ that is a left [[module]] over a ring $R$ and a right module over a ring $S$, and is such that $(rb)s = r(bs)$ for all $r\in R$, $ b \in B$, $s \in S$. One writes ${}_R B_S$, or that $B$ is an $(R,S)$-bimodule. The bimodule $B$ may be regarded as a left $R \otimes S^{\mathrm{op}}$-module, where $S^{\mathrm{op}}$ is the [[opposite ring]] (dually isomorphic, anti-isomorphic) to $S$, while $\otimes$ denotes the tensor product over the ring of integers, and $(r\otimes s)b = rbs$. For every left $R$-module $M$ one has the situation ${}_R M_E$, where $E$ is the ring of endomorphisms of $M$. Any ring $R$ can be given the natural structure of an $(R,R)$-bimodule. |
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| | ====Comments==== | | ====Comments==== |
| − | A bimodule morphism is a mapping from a bimodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631024.png" /> into a bimodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631025.png" /> that is left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631026.png" />-linear and right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631027.png" />-linear. The category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631028.png" />-bimodules with bimodule morphisms is a [[Grothendieck category|Grothendieck category]]. | + | A bimodule morphism is a mapping from a bimodule ${}_R B_S$ into a bimodule ${}_R C_S$ that is left $R$-linear and right $S$-linear. The category of $(R,S)$-bimodules with bimodule morphisms is a [[Grothendieck category]]. |
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| | + | The centre of an $(R,R)$-bimodule (also called an $R$-bimodule) $B$ is defined to be the set |
| | + | $$ |
| | + | Z_R(B) = \{x \in B : rx = xr \ \text{for all}\ r \in R\}\ . |
| | + | $$ |
| | + | Clearly $Z_R(B)$ is a two-sided $Z_R(R)$--module. In particular, when $R$ is commutative, the distinction between left and right modules disappears and any $R$-module may be regarded as an $(R,R)$-bimodule. |
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| − | The centre of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631029.png" />-bimodule (also called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631030.png" />-bimodule) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631031.png" /> is defined to be the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631032.png" />. Clearly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631033.png" /> is a two-sided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016310/b01631034.png" />-module.
| + | ====References==== |
| | + | <table> |
| | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> Paul M. Cohn. ''Basic Algebra: Groups, Rings, and Fields'', Springer (2003) {{ISBN|1852335874}}. {{ZBL|1003.00001}}</TD></TR> |
| | + | </table> |
Latest revision as of 17:01, 23 November 2023
2020 Mathematics Subject Classification: Primary: 16D20 [MSN][ZBL]
double module
An Abelian group $B$ that is a left module over a ring $R$ and a right module over a ring $S$, and is such that $(rb)s = r(bs)$ for all $r\in R$, $ b \in B$, $s \in S$. One writes ${}_R B_S$, or that $B$ is an $(R,S)$-bimodule. The bimodule $B$ may be regarded as a left $R \otimes S^{\mathrm{op}}$-module, where $S^{\mathrm{op}}$ is the opposite ring (dually isomorphic, anti-isomorphic) to $S$, while $\otimes$ denotes the tensor product over the ring of integers, and $(r\otimes s)b = rbs$. For every left $R$-module $M$ one has the situation ${}_R M_E$, where $E$ is the ring of endomorphisms of $M$. Any ring $R$ can be given the natural structure of an $(R,R)$-bimodule.
A bimodule morphism is a mapping from a bimodule ${}_R B_S$ into a bimodule ${}_R C_S$ that is left $R$-linear and right $S$-linear. The category of $(R,S)$-bimodules with bimodule morphisms is a Grothendieck category.
The centre of an $(R,R)$-bimodule (also called an $R$-bimodule) $B$ is defined to be the set
$$
Z_R(B) = \{x \in B : rx = xr \ \text{for all}\ r \in R\}\ .
$$
Clearly $Z_R(B)$ is a two-sided $Z_R(R)$--module. In particular, when $R$ is commutative, the distinction between left and right modules disappears and any $R$-module may be regarded as an $(R,R)$-bimodule.
References
| [a1] | Paul M. Cohn. Basic Algebra: Groups, Rings, and Fields, Springer (2003) ISBN 1852335874. Zbl 1003.00001 |
How to Cite This Entry:
Bimodule. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bimodule&oldid=35237
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article