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''(left), left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081840/r0818402.png" />-ring''
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''(left), left $RR$-ring''
  
A ring in which the left [[Annihilator|annihilator]] of any element is generated by an [[Idempotent|idempotent]] (right Rickart rings are defined in a symmetric way). Rickart rings are characterized by the projectivity of all principal left (right) ideals. Regular, Baer and semi-hereditary rings are Rickart rings. A left Rickart ring need not be a right Rickart ring. A ring of matrices over a Rickart ring need not be a Rickart ring. The ring of endomorphisms of all free left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081840/r0818403.png" />-modules is a Rickart ring if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081840/r0818404.png" /> is left-hereditary. All these rings are right Rickart rings if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081840/r0818405.png" /> is left-hereditary, left-perfect and right-coherent. Under these conditions the rings of endomorphisms prove to be Baer rings (see [[Regular ring (in the sense of von Neumann)|Regular ring (in the sense of von Neumann)]]). A commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081840/r0818406.png" /> is a Rickart ring if and only if its full ring of fractions is regular in the sense of von Neumann and if for every maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081840/r0818407.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081840/r0818408.png" /> the ring of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081840/r0818409.png" /> does not have zero divisors. A ring of polynomials over a commutative Rickart ring is a Rickart ring.
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A ring in which the left [[annihilator]] of any element is generated by an [[idempotent]] (right Rickart rings are defined in a symmetric way). Rickart rings are characterized by the projectivity of all principal left (right) ideals. Regular, Baer and semi-hereditary rings are Rickart rings. A left Rickart ring need not be a right Rickart ring. A ring of matrices over a Rickart ring need not be a Rickart ring. The ring of endomorphisms of all free left $R$-modules is a Rickart ring if and only if $R$ is left-hereditary. All these rings are right Rickart rings if and only if $R$ is left-hereditary, left-perfect and right-coherent. Under these conditions the rings of endomorphisms prove to be Baer rings (see [[Regular ring (in the sense of von Neumann)]]). A commutative ring $R$ is a Rickart ring if and only if its full ring of fractions is regular in the sense of von Neumann and if for every maximal ideal $\mathfrak{M}$ of $R$ the ring of fractions $R_{\mathfrak{M}}$ does not have zero divisors. A ring of polynomials over a commutative Rickart ring is a Rickart ring.
  
A ring with an involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081840/r08184010.png" /> is called a Rickart <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081840/r08184012.png" />-ring if the left annihilator of any element is generated by a projection, i.e. by an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081840/r08184013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081840/r08184014.png" />. The analogous property for right annihilators is automatically fulfilled in this case. The projections of a Rickart <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081840/r08184015.png" />-ring form a lattice. This lattice is complete if and only if the annihilator of any set is generated by projections. Such rings are known as Baer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081840/r08184017.png" />-rings. The term  "Rickart ring"  was introduced in honour of C.E. Rickart, who studied the corresponding property in rings of operators (see [[#References|[1]]]).
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A ring with an involution $*$ is called a Rickart $*$-ring if the left annihilator of any element is generated by a projection, i.e. by an element $e$ such that $e = e^2 = e^*$. The analogous property for right annihilators is automatically fulfilled in this case. The projections of a Rickart $*$-ring form a [[lattice]]. This is a [[complete lattice]] if and only if the annihilator of any set is generated by projections. Such rings are known as Baer $*$-rings. The term  "Rickart ring"  was introduced in honour of C.E. Rickart, who studied the corresponding property in rings of operators (see [[#References|[1]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.E. Rickart,  "Banach algebras with an adjoint operation"  ''Ann. of Math.'' , '''47'''  (1946)  pp. 528–550</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.K. Berberian,  "Baer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081840/r08184018.png" />-rings" , Springer  (1972)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I. Kaplansky,  "Rings of operators" , Benjamin  (1968)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.T. Markov,  A.V. Mikhalev,  L.A. Skornyakov,  A.A. Tuganbaev,  "Modules"  ''J. Soviet Math.'' , '''23''' :  6  (1983)  pp. 2642–2707  ''Itogi Nauk. i Tekn. Algebra Topol. Geom.'' , '''19'''  (1981)  pp. 31–134</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  C.E. Rickart,  "Banach algebras with an adjoint operation"  ''Ann. of Math.'' , '''47'''  (1946)  pp. 528–550</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  S.K. Berberian,  "Baer $*$-rings" , Springer  (1972)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  I. Kaplansky,  "Rings of operators" , Benjamin  (1968)</TD></TR>
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<TR><TD valign="top">[4]</TD> <TD valign="top">  V.T. Markov,  A.V. Mikhalev,  L.A. Skornyakov,  A.A. Tuganbaev,  "Modules"  ''J. Soviet Math.'' , '''23''' :  6  (1983)  pp. 2642–2707  ''Itogi Nauk. i Tekn. Algebra Topol. Geom.'' , '''19'''  (1981)  pp. 31–134</TD></TR>
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Revision as of 20:50, 7 October 2017

(left), left $RR$-ring

A ring in which the left annihilator of any element is generated by an idempotent (right Rickart rings are defined in a symmetric way). Rickart rings are characterized by the projectivity of all principal left (right) ideals. Regular, Baer and semi-hereditary rings are Rickart rings. A left Rickart ring need not be a right Rickart ring. A ring of matrices over a Rickart ring need not be a Rickart ring. The ring of endomorphisms of all free left $R$-modules is a Rickart ring if and only if $R$ is left-hereditary. All these rings are right Rickart rings if and only if $R$ is left-hereditary, left-perfect and right-coherent. Under these conditions the rings of endomorphisms prove to be Baer rings (see Regular ring (in the sense of von Neumann)). A commutative ring $R$ is a Rickart ring if and only if its full ring of fractions is regular in the sense of von Neumann and if for every maximal ideal $\mathfrak{M}$ of $R$ the ring of fractions $R_{\mathfrak{M}}$ does not have zero divisors. A ring of polynomials over a commutative Rickart ring is a Rickart ring.

A ring with an involution $*$ is called a Rickart $*$-ring if the left annihilator of any element is generated by a projection, i.e. by an element $e$ such that $e = e^2 = e^*$. The analogous property for right annihilators is automatically fulfilled in this case. The projections of a Rickart $*$-ring form a lattice. This is a complete lattice if and only if the annihilator of any set is generated by projections. Such rings are known as Baer $*$-rings. The term "Rickart ring" was introduced in honour of C.E. Rickart, who studied the corresponding property in rings of operators (see [1]).

References

[1] C.E. Rickart, "Banach algebras with an adjoint operation" Ann. of Math. , 47 (1946) pp. 528–550
[2] S.K. Berberian, "Baer $*$-rings" , Springer (1972)
[3] I. Kaplansky, "Rings of operators" , Benjamin (1968)
[4] V.T. Markov, A.V. Mikhalev, L.A. Skornyakov, A.A. Tuganbaev, "Modules" J. Soviet Math. , 23 : 6 (1983) pp. 2642–2707 Itogi Nauk. i Tekn. Algebra Topol. Geom. , 19 (1981) pp. 31–134
How to Cite This Entry:
Rickart ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rickart_ring&oldid=42034
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article