''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
2 KB (265 words) - 16:14, 11 September 2016
...s, all elements of an Engel algebra are [[Engel element]]s, see also [[Lie algebra, nil]]).
325 bytes (51 words) - 18:39, 16 October 2016
''right Artinian ring''
...mber $m$ such that $B_m=B_{m+1}=\dotsb$. The definition of a left Artinian ring is similar.
3 KB (460 words) - 12:07, 14 February 2020
The same as a normed or [[Banach algebra|Banach algebra]].
A ring with a [[Valuation|valuation]] on it.
104 bytes (18 words) - 17:26, 7 February 2011
...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]].
2 KB (318 words) - 09:55, 26 November 2016
...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
2 KB (335 words) - 15:09, 13 August 2014
...$k$ 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> <
2 KB (341 words) - 12:10, 25 August 2014
...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
1 KB (163 words) - 08:20, 12 November 2023
$#A+1 = 44 n = 0
$#C+1 = 44 : ~/encyclopedia/old_files/data/R082/R.0802450 Ring with operators,
4 KB (634 words) - 07:02, 30 March 2024
...hat R)$ 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
2 KB (291 words) - 07:26, 29 November 2014
====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
2 KB (316 words) - 21:57, 29 November 2014
...[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>
1 KB (157 words) - 11:43, 29 June 2014
...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
3 KB (403 words) - 17:47, 27 August 2014
...of finite type over a Bezout ring is a direct sum of a torsion module and a free module.
1 KB (190 words) - 19:52, 2 November 2014
...lar, associative) ring $R$ is called quasi-regular if there is an element $a'\in R$ such that
$$a+a'+aa'=a+a'+a'a=0.$$
1 KB (189 words) - 21:27, 6 September 2017
...= 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.
2 KB (305 words) - 16:09, 11 September 2016
''of a ring $R$''
The ring $R^{\mathrm{op}}$ having the same additive group as $R$ but with multiplica
345 bytes (55 words) - 11:57, 23 November 2023
''over a field $F$''
...over a field $F$ that generalises the construction of the [[quaternion]]s over the field of real numbers.
1 KB (180 words) - 20:49, 5 December 2023
...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]]].
2 KB (327 words) - 07:32, 13 December 2016
...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>
1 KB (161 words) - 08:20, 30 November 2014