Difference between revisions of "Rickart ring"
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''(left), left $RR$-ring'' | ''(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 ring (in the sense of von Neumann)|Regular (in the sense of von Neumann)]], Baer and [[semi-hereditary ring]]s 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 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 ring (in the sense of von Neumann)|Regular (in the sense of von Neumann) rings]], [[Baer ring]]s and [[semi-hereditary ring]]s 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 [[#References|[1]]]). | 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]]]). | ||
Latest revision as of 06:38, 18 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 (in the sense of von Neumann) rings, Baer rings 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 |
Rickart ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rickart_ring&oldid=42035