...lly in the meaning of [[Von Neumann algebra|von Neumann algebra]] or $W^*$-algebra.
2) A [[Ring with operators|ring with operators]].
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...mple of an non-associative commutative ring is a Jordan ring (see [[Jordan algebra]]).
[[Category:Algebra]]
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...e algebra over a field possessing a unit element is imbeddable in a simple algebra with the same unit element.
For references, see [[Simple ring]].
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...ar-algebra(2)|linear algebra]], [[multilinear algebra]], and [[topological algebra]].
An [[algebraic system]], consisting of a set with a family of algebraic operations.
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The centre of a ring $R$ is
the collection $Z$ of all elements of the ring $R$ that commute with every
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''unitary ring, ring with identity, ring-with-a-one''
...ltiplication $(m,r) \cdot (n,s) = (mn, ms + nr + rs)$ which has $(1,0)$ as a multiplicative identity.
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...erfect]]. A [[Noetherian ring|Noetherian]] balanced ring is an [[Artinian ring]].
<TR><TD valign="top">[1]</TD> <TD valign="top"> ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''19''' (1981) pp. 31–134</TD></TR>
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...ing|ring]] which is a left (right) [[Semi-chain module|semi-chain module]] over itself.
...e of a serial ring is the ring of formal power series in one variable over a field.
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...trass rings are [[analytic ring]]s (rings of convergent power series) over a perfect field, to which the [[Weierstrass preparation theorem]] is applicab
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''with respect to a radical $r$''
...ebras]]; [[Jordan algebra|Jordan algebra]]; [[Lie algebra, semi-simple|Lie algebra, semi-simple]]).
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A ring with division is a (not necessarily associative) [[ring]] in which the equations
$$a \cdot x = b \quad;\quad\quad y \cdot a = b $$
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...operator ring]], when the domain of operators is a commutative associative ring with unit.
...onal vector spaces and linear operators between them. See the entry Linear algebra below.
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...acobson ring. On the contrary, a local non-Artinian ring is not a Jacobson ring.
...o the spectrum $\mathrm{Spec}(A)$; this definition leads to the concept of a Jacobson scheme.
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...full matrix ring of a [[local ring]]. This holds, in particular, if $J$ is a [[nil ideal]].
<TR><TD valign="top">[2]</TD> <TD valign="top"> C. Faith, "Algebra" , '''1–2''' , Springer (1973–1976)</TD></TR>
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...e additive group of $B$ and for which $(ab)\phi=b\phi\cdot a\phi$ ($a,b\in A$).
The [[opposite ring]] $R^{\mathrm{op}}$ to a ring $R$ is anti-isomorphic to $R$.
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...e. $\partial(\alpha x) = \alpha \partial(x)$, where $\alpha \in K$, $x \in A$.
Cf. also [[Derivation in a ring]].
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''$M$ over a commutative ring $R$''
.... For example, for a finite Abelian group$M$ regarded as a module over the ring of integers, $\mathrm{Supp}(M)$ consists of all prime ideals $(p)$, where $
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The term ''regular algebra'' may refer to:
* a [[regular ring (in commutative algebra)]];
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...he first one is. Every automorphism of a finite-dimensional central simple algebra is inner and its dimension is the square of an integer.
...="top">[2]</TD> <TD valign="top"> L.A. Skornyakov, "Elements of general algebra" , Moscow (1983) (In Russian)</TD></TR>
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...o be strongly degenerate. The smallest ideal of a Lie algebra the quotient algebra by which is not strongly degenerate is called the Kostrikin radical.
...60 (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.I. Kostrikin, "Around Burnside" , Springer (1989) (Translated from Russ
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''integral ring''
..., is an integral domain. Conversely, an integral domain can be imbedded in a field. Such an imbedding is given by the construction of the [[field of fra
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...s, all elements of an Engel algebra are [[Engel element]]s, see also [[Lie algebra, nil]]).
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''right Artinian ring''
...mber $m$ such that $B_m=B_{m+1}=\dotsb$. The definition of a left Artinian ring is similar.
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The same as a normed or [[Banach algebra|Banach algebra]].
A ring with a [[Valuation|valuation]] on it.
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...simple rings, cf. also [[Noetherian ring|Noetherian ring]]) different from a skew-field, as well as Noetherian simple rings with zero divisors but witho
...gebras|Alternative rings and algebras]]). See also [[Simple algebra|Simple algebra]].
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...an ring $K\neq0$ is a direct sum of fields $\mathbf Z_2$ and therefore has a unit element.
...ebra|Boolean algebra]], namely: Any Boolean algebra is a Boolean ring with a unit element under the operations of addition and multiplication defined by
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...$k$ is a perfect field, an analytic ring is an [[Excellent ring|excellent ring]].
...ntiels de régularité pour les localisés des algèbres analytiques" ''J. of Algebra'' , '''5''' (1967) pp. 305–324</TD></TR><TR><TD valign="top">[2]</TD> <
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...that a non-null subalgebra of a free algebra over a field $\Phi$ is a free algebra.
* Kurosh, A. "Nonassociative free algebras and free products of algebras". Mat. Sb., N
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$#A+1 = 44 n = 0
$#C+1 = 44 : ~/encyclopedia/old_files/data/R082/R.0802450 Ring with operators,
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...hat R)$ is the endomorphism ring of the right $R$-module $\widehat R$. The ring $Q_{\mathrm{max}}(R)$ can also be defined as the [[direct limit]]
...f all dense right ideals of $R$ (a right ideal $D$ of a ring $R$ is called a dense ideal if
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====Separable algebra over a field====
...$ is separable if and only if the centres of the simple components of this algebra (see [[Associative rings and algebras|Associative rings and algebras]]) are
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...[2]]] established that any [[Noetherian ring|Noetherian ring]] is a Lasker ring.
...D> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</TD></TR></table>
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...irect sum of modules dual to finitely-generated right ideals of the ground ring.
...are projective. The ring of polynomials in one variable over a commutative ring $T$ is semi-hereditary if and only if $R$ is regular in the sense of von Ne
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...of finite type over a Bezout ring is a direct sum of a torsion module and a free module.
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...lar, associative) ring $R$ is called quasi-regular if there is an element $a'\in R$ such that
$$a+a'+aa'=a+a'+a'a=0.$$
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...= R$, then $R$ is said to be integrally closed in $S$ (cf. also [[Integral ring]]).
...nd is closed in its field of fractions. In some of the literature a normal ring is also required to be an integral domain.
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''of a ring $R$''
The ring $R^{\mathrm{op}}$ having the same additive group as $R$ but with multiplica
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''over a field $F$''
...over a field $F$ that generalises the construction of the [[quaternion]]s over the field of real numbers.
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...om the properties of the category of all left (or right) modules over this ring (cf. [[Morita equivalence]]; [[Module]]).
...their projectivity, and also to the injectivity of all left ideals of the ring [[#References|[1]]].
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...determines a [[closed subscheme]] of the [[affine scheme]] $\mathrm{Spec}\,A$, consisting of the points that are not normal.
...D> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</TD></TR>
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...ing has a left classical ring of fractions that is a classical semi-simple ring if and only if it satisfies the maximum condition for left annihilators and
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A collection $\mathcal{A}$ of subsets of a set $X$ satisfying:
i) $\emptyset\in \mathcal{A}$;
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...r matrices over a field the matrices with 0's along the main diagonal form a nilpotent ideal.
...in a [[Noetherian ring|Noetherian ring]]. In a left (or right) Noetherian ring every left (right) nil ideal is nilpotent.
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...es the ring of integers and all complete local rings. See also [[Excellent ring]].
...="top">[1]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''1''' , Springer (1975)</TD></TR>
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...tive ring $R$ is said to be non-full if it can be represented in the form $A=BC$ where $B$, $C$ are matrices of orders $n\times r$ and $r\times n$, resp
$$A=(a,a_2,\ldots,a_n),\quad B=(b,a_2,\ldots,a_n)$$
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...and algebras]]). Right perfect rings are defined similarly. A left perfect ring need not be right perfect.
...-module has a decomposition with respect to which every direct summand has a complement (see [[Krull–Remak–Schmidt theorem|Krull–Remak–Schmidt t
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''(left), left $RR$-ring''
...visors. A ring of polynomials over a commutative Rickart ring is a Rickart ring.
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$#A+1 = 6 n = 2
...ive module]]). The Prüfer domains are the commutative integral domains $ A $
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$#A+1 = 30 n = 0
...~/encyclopedia/old_files/data/R080/R.0800820 Regular ring (in commutative algebra)
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''of a ring $R$ by an ideal $I$''
$$(a+I)(b+I)=ab+I.$$
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