View source for Cross product

''crossed product, of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c0271201.png" /> and a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c0271202.png" />''

An associative ring defined as follows. Suppose one is given a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c0271203.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c0271204.png" /> into the isomorphism group of an associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c0271205.png" /> with an identity, and a family

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of invertible elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c0271207.png" />, satisfying the conditions

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c0271209.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712011.png" />. The family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712012.png" /> is called a factor system. Then the cross product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712014.png" /> with respect to the factor system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712015.png" /> and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712016.png" /> is the set of all formal finite sums of the form

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(where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712018.png" /> are symbols uniquely assigned to every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712019.png" />), with binary operations defined by

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This ring is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712022.png" />; the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712023.png" /> form a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712024.png" />-basis of it.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712025.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712026.png" /> onto the identity automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712027.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712028.png" /> is called a twisted or crossed group ring, and if, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712029.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712030.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712031.png" /> is the group ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712032.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712033.png" /> (see [[Group algebra|Group algebra]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712034.png" /> be a field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712035.png" /> a monomorphism. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712036.png" /> is a simple ring, being the cross product of the field with its Galois group.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.K. Sehgal,   "Topics in group rings" , M. Dekker  (1978)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Bovdi,   "Cross products of semi-groups and rings"  ''Sibirsk. Mat. Zh.'' , '''4'''  (1963)  pp. 481–499  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.E. Zalesskii,   A.V. Mikhalev,   "Group rings"  ''J. Soviet Math.'' , '''4'''  (1975)  pp. 1–74  ''Itogi Nauk. i Tekhn. Sovrem. Probl. Mat.'' , '''2'''  (1973)  pp. 5–118</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D.S. Passman,   "The algebraic structure of group rings" , Wiley  (1977)</TD></TR></table>



====Comments====
In the defining relations for a factor system above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712037.png" />, e.g., of course stands for the result of applying the automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712038.png" /> to the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712039.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712040.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712041.png" />, then one obtains the skew group ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712042.png" />. Cross products arise naturally when dealing with extensions. Indeed, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712043.png" /> be a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712044.png" />. Choose a set of representatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712045.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712046.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712047.png" />. Then every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712048.png" />, the group algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712049.png" />, can be written as a unique sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712051.png" />. Now write

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Then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712053.png" /> define a factor system (for the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712054.png" /> and the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712055.png" /> relative to the set of automorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712056.png" />) and

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Up to Brauer equivalence every [[Central simple algebra|central simple algebra]] is a cross product, but not every [[Division algebra|division algebra]] is isomorphic to a cross product. Two algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712058.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712059.png" /> are Brauer equivalent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712060.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712061.png" /> for suitable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712063.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712064.png" /> is the algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712065.png" /> matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027120/c02712066.png" />. Cf. also [[Brauer group|Brauer group]].