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The representation of a ring as the direct sum of subrings related to a given idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p0719701.png" />. For a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p0719702.png" /> containing an idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p0719703.png" />, there exist left, right and two-sided Peirce decompositions, which are defined by
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p0719701.png
 
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$#A+1 = 51 n = 0
 
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$#C+1 = 51 : ~/encyclopedia/old_files/data/P071/P.0701970 Peirce decomposition
 
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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/p/p071/p071970/p0719704.png" /></td> </tr></table>
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The representation of a ring as the direct sum of subrings related to a given idempotent  $  e $.  
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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/p/p071/p071970/p0719705.png" /></td> </tr></table>
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For a ring  $  R $
 
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containing an idempotent  $  e $,
 
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there exist left, right and two-sided Peirce decompositions, which are defined by
 
  
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$$
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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/p/p071/p071970/p0719706.png" /></td> </tr></table>
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R  =   \mathop{\rm Re} + R( 1- e),
 
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$$
 
  
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$$
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respectively. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p0719707.png" /> has no identity, then one puts, by definition,
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R  = eR + ( 1- e) R,
 
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$$
 
  
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$$
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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/p/p071/p071970/p0719708.png" /></td> </tr></table>
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R  = eRe + eR( 1- e)+( 1- e) Re+( 1- e) R( 1- e),
 
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$$
 
  
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respectively. If  $  R $
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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/p/p071/p071970/p0719709.png" /></td> </tr></table>
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has no identity, then one puts, by definition,
 
  
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$$
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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/p/p071/p071970/p07197010.png" /></td> </tr></table>
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R( 1- e)  = \{ {x- xe } : {x \in R } \}
 
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,
 
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$$
 
  
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$$
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The sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197012.png" /> are defined analogously. Therefore, in a two-sided Peirce decomposition an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197013.png" /> can be represented as
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( 1- e) Re  = \{ xe- exe: x \in R \} ,
 
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$$
 
  
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$$
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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/p/p071/p071970/p07197014.png" /></td> </tr></table>
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( 1- e) R( 1- e)  = \{ x- ex- xe+ exe: x \in R \} .
 
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$$
 
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The sets  $  ( 1- e) R $
 
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and  $  eR( 1- e) $
 
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are defined analogously. Therefore, in a two-sided Peirce decomposition an element  $  x \in R $
 
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can be represented as
 
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$$
 
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x  =  exe+( ex- exe)+( xe- exe)+( x- ex- xe+ exe),
 
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$$
 
  
 
in a left decomposition as
 
in a left decomposition as
  
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$$
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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/p/p071/p071970/p07197015.png" /></td> </tr></table>
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x  = xe+( x- xe) ,
 
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$$
 
  
 
and in a right decomposition as
 
and in a right decomposition as
  
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$$
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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/p/p071/p071970/p07197016.png" /></td> </tr></table>
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x  = ex +( x- ex).
 
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$$
 
  
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There is also a Peirce decomposition with respect to an orthogonal system of idempotents $  \{ e _ {1} \dots e _ {n} \} $
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There is also a Peirce decomposition with respect to an orthogonal system of idempotents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197017.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197018.png" />:
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where $  \sum _ {i} e _ {i} = 1 $:
 
  
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$$
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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/p/p071/p071970/p07197019.png" /></td> </tr></table>
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R  = \sum _ {i,j } e _ {i}  \mathop{\rm Re} _ {j} .
 
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$$
 
  
 
This decomposition was proposed by B. Peirce [[#References|[1]]].
 
This decomposition was proposed by B. Peirce [[#References|[1]]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Peirce,  "Linear associative algebra"  ''Amer. J. Math.'' , '''4'''  (1881)  pp. 97–229</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Peirce,  "Linear associative algebra"  ''Amer. J. Math.'' , '''4'''  (1881)  pp. 97–229</TD></TR></table>
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 +
  
 
====Comments====
 
====Comments====
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In modern ring theory the Peirce decomposition appears in the ring of a Morita context $  ( R, S, V, W) $,  
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In modern ring theory the Peirce decomposition appears in the ring of a Morita context <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197022.png" /> are Morita related if they are subrings of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197023.png" /> with an idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197026.png" />, i.e., they are parts of a Peirce decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197027.png" /> (see [[#References|[a3]]], p.12).
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where $  R $
 
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and $  S $
 
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are Morita related if they are subrings of a ring $  T $
 
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with an idempotent $  e $
 
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such that $  R= eTe $,  
 
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$  S=( 1- e) T( 1- e) $,  
 
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i.e., they are parts of a Peirce decomposition of $  T $(
 
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see [[#References|[a3]]], p.12).
 
  
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A context or a set of pre-equivalence data is a sextuple $  ( R, R  ^  \prime  , M , M  ^  \prime  , \tau , \tau  ^  \prime  ) $
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A context or a set of pre-equivalence data is a sextuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197028.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197030.png" /> are rings, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197031.png" /> is a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197032.png" />-, right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197033.png" />-bimodule, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197034.png" /> is a right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197035.png" />-, left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197036.png" />-bimodule and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197038.png" /> are bimodule homomorphisms, such that the following two associativity diagrams commute:
−
where $  R $
 
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and $  R  ^  \prime  $
 
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are rings, $  M $
 
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is a left $  R $-,  
 
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right $  R  ^  \prime  $-
 
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bimodule, $  M  ^  \prime  $
 
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is a right $  R $-,  
 
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left $  R  ^  \prime  $-
 
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bimodule and $  \tau : M \otimes _ {R  ^  \prime  } M  ^  \prime  \rightarrow R $,  
 
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$  \tau  ^  \prime  : M  ^  \prime  \otimes _ {R} M \rightarrow R  ^  \prime  $
 
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are bimodule homomorphisms, such that the following two associativity diagrams commute:
 
  
−
$$
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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/p/p071/p071970/p07197039.png" /></td> </tr></table>
  
 
and
 
and
  
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$$
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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/p/p071/p071970/p07197040.png" /></td> </tr></table>
  
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Using $  \tau , \tau  ^  \prime  $,
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Using <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197041.png" />, the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197042.png" />-matrices
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the set of all $  ( 2 \times 2) $-
 
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matrices
 
  
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$$
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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/p/p071/p071970/p07197043.png" /></td> </tr></table>
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\left (
 
  
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$$
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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/p/p071/p071970/p07197044.png" /></td> </tr></table>
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= \
 
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\left \{ \left (
 
  
 
acquires a multiplication (using the usual matrix formulas) and this multiplication is associative precisely if the two diagrams above commute. Such a ring is then called the ring of a Morita context.
 
acquires a multiplication (using the usual matrix formulas) and this multiplication is associative precisely if the two diagrams above commute. Such a ring is then called the ring of a Morita context.
  
−
If $  ( R, R  ^  \prime  , M, M  ^  \prime  , \tau , \tau  ^  \prime  ) $
+
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197045.png" /> is a Morita context with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197047.png" /> epic, then the functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197049.png" /> define an equivalence of categories between the categories of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197050.png" />-modules and right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071970/p07197051.png" />-modules; cf. also [[Morita equivalence|Morita equivalence]]. Cf. [[#References|[a1]]], §4.1 for more details.
−
is a Morita context with $  \tau $
 
−
and $  \tau  ^  \prime  $
 
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epic, then the functors $  N \mapsto M  ^  \prime  \otimes _ {R} N $,  
 
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$  N  ^  \prime  \mapsto M \otimes _ {R  ^  \prime  } N  ^  \prime  $
 
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define an equivalence of categories between the categories of left $  R $-
 
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modules and right $  R  ^  \prime  $-
 
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modules; cf. also [[Morita equivalence|Morita equivalence]]. Cf. [[#References|[a1]]], §4.1 for more details.
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.H. Rowen,  "Ring theory" , '''I''' , Acad. Press  (1988)  pp. 36</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)  pp. 48, 50</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.C. McConnell,  J.C. Robson,  "Noncommutative Noetherian rings" , Wiley  (1987)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.H. Rowen,  "Ring theory" , '''I''' , Acad. Press  (1988)  pp. 36</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)  pp. 48, 50</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.C. McConnell,  J.C. Robson,  "Noncommutative Noetherian rings" , Wiley  (1987)</TD></TR></table>

Revision as of 14:52, 7 June 2020

The representation of a ring as the direct sum of subrings related to a given idempotent . For a ring containing an idempotent , there exist left, right and two-sided Peirce decompositions, which are defined by

respectively. If has no identity, then one puts, by definition,

The sets and are defined analogously. Therefore, in a two-sided Peirce decomposition an element can be represented as

in a left decomposition as

and in a right decomposition as

There is also a Peirce decomposition with respect to an orthogonal system of idempotents where :

This decomposition was proposed by B. Peirce [1].

References

[1] B. Peirce, "Linear associative algebra" Amer. J. Math. , 4 (1881) pp. 97–229


Comments

In modern ring theory the Peirce decomposition appears in the ring of a Morita context , where and are Morita related if they are subrings of a ring with an idempotent such that , , i.e., they are parts of a Peirce decomposition of (see [a3], p.12).

A context or a set of pre-equivalence data is a sextuple where and are rings, is a left -, right -bimodule, is a right -, left -bimodule and , are bimodule homomorphisms, such that the following two associativity diagrams commute:

and

Using , the set of all -matrices

acquires a multiplication (using the usual matrix formulas) and this multiplication is associative precisely if the two diagrams above commute. Such a ring is then called the ring of a Morita context.

If is a Morita context with and epic, then the functors , define an equivalence of categories between the categories of left -modules and right -modules; cf. also Morita equivalence. Cf. [a1], §4.1 for more details.

References

[a1] L.H. Rowen, "Ring theory" , I , Acad. Press (1988) pp. 36
[a2] N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) pp. 48, 50
[a3] J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987)
How to Cite This Entry:
Peirce decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Peirce_decomposition&oldid=49362
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Peirce_decomposition&oldid=49362"