Added or improved some keyword descriptions

db/ringapp/keyword/KWD_000001.yaml

@@ -1,2 +1,2 @@
-description: null
+description: The ring of fractions with numerators and denominators from a commutative <a href="keywords/keyword/2/">polynomial ring</a> (possibly multivariate). Often called the ring of rational functions.
 name: rational polynomial ring

db/ringapp/keyword/KWD_000002.yaml

@@ -1,2 +1,2 @@
-description: null
+description: The polynomial ring $R[x]$ is the <a href="keywords/keyword/35/">monoid ring</a> of the monoid $\Bbb N$ of nonnegative integers over a ring $R$. The polynomial ring $R[X]$, where $X$ is a set of variables (possibly infinite), is the monoid ring of the direct sum $\bigoplus\limits_{x \in X} \Bbb N$.
 name: polynomial ring

db/ringapp/keyword/KWD_000003.yaml

@@ -1,2 +1,2 @@
-description: null
+description: The set of residue classes of elements of a ring $R$ modulo a two-sided ideal $R$ with operations induced from those of $R$. Not to be confused with <a href="keywords/keyword/24/">rings of quotients</a>.
 name: quotient ring

db/ringapp/keyword/KWD_000005.yaml

@@ -1,2 +1,2 @@
-description: null
+description: The localization $S^{-1} R$ of a commutative ring $R$ by a multiplicatively closed set $S$ is defined as the set of pairs $(r,s)$ (written as $r/s$), where $r/s = r'/s'$ if and only if there exists $t\in S$ with $t(r/s-r'/s')=0$. (If $R$ is an integral domain, the natural map $R \to S^{-1} R$, $r \mapsto r/1$, is an embedding.) There are multiple generalizations of this process to noncommutative rings, often referred to as (noncommutative) localization; the resulting rings are called <a href="keywords/keyword/24/">rings of quotients</a>.
 name: localization

db/ringapp/keyword/KWD_000008.yaml

@@ -1,2 +1,2 @@
-description: null
+description: A ring of matrices whose columns and rows are indexed by an infinite set.
 name: infinite matrix ring

db/ringapp/keyword/KWD_000009.yaml

@@ -1,4 +1,4 @@
-description: 'Given a commutative ring $R$ and an $R$ module $M$, the underlying set
+description: 'Given a ring $R$ and an $R$-bimodule $M$, the underlying set
   is $R\times M$. Addition: $(r,m)+(r'',m'')=(r+r'', m+m'')$. Multiplication: $(r,m)(r'',m'')=(rr'',
   rm''+mr'')$'
 name: trivial extension

db/ringapp/keyword/KWD_000010.yaml

@@ -1,4 +1,2 @@
-description: Given a ring $R$ and a semigroup $S$, the underlying set is the set of
-  finite $R$-combinations of elements of $S$. Addition and multiplication defined
-  the same as for group rings.
+description: Given a ring $R$ and a semigroup $S$ (possibly noncommutative), the semigroup ring $R[S]$ is the set of finite formal $R$-combinations of elements of $S$ with pointwise addition $(\sum_g a_gg)(\sum_h b_hh)=\sum_k (a_k+b_k)k$ and convolutional multiplication $(\sum_g a_gg)(\sum_h b_hh):=\sum_{gh=k}a_gb_hk$. (If $S$ is not a monoid, $R[S]$ is a ring without identity.)
 name: semigroup ring

db/ringapp/keyword/KWD_000013.yaml

@@ -1,5 +1,5 @@
 description: Given a ring $R$ and an indeterminate $X$, the underlying set is the
   set of infinite formal $R$-combinations of $\{X^i\mid i\in \mathbb Z\}$, but only
   finitely many negative powers of $X$ can have nonzero coefficients. Addition and
-  multiplication as with polynomials and power series.
+  multiplication are as with polynomials and power series.
 name: Laurent series ring

db/ringapp/keyword/KWD_000014.yaml

@@ -1,4 +1,4 @@
 description: For a ring $R$ and an indeterminate $X$, the underlying set is the set
   of all finite $R$-combinations of $\{X^i\mid i\in \mathbb Z\}$. Addition and multiplication
-  just as with polynomial multiplication.
+  are just as with polynomial multiplication.
 name: Laurent polynomials

db/ringapp/keyword/KWD_000018.yaml

@@ -1,2 +1,2 @@
-description: null
+description: Let $A \subset B$ be commutative rings. The integral closure of $A$ in $B$ is the subring of elements of $B$ integral over $A$, that is, of elements that satisfy monic polynomial equations with coefficients in $A$.
 name: integral closure

db/ringapp/keyword/KWD_000019.yaml

@@ -1,2 +1,2 @@
-description: null
+description: "Let $\delta$: $R \to R$ be a derivation (an Abelian group homomorphism that satisfies $\delta(ab)=a\delta(b)+\delta(a)b$). Then the rule $xa=ax+\delta(a)$ defines a ring structure on the set of left polynomials $\sum a_i x^i$ ($a_i \in R$), the ring is denoted R[x,\delta]."
 name: differential polynomial ring

db/ringapp/keyword/KWD_000021.yaml

@@ -1,4 +1,2 @@
-description: 'Given a group $G$ and a ring $R$, the underlying set of $R[G]$ is the
-  set of finite linear combinations using $G$ as a basis.  Addition: $(\sum_g a_gg)(\sum_h
-  b_hh)=\sum_k (a_k+b_k)k$.  Multiplication: $(\sum_g a_gg)(\sum_h b_hh):=\sum_{gh=k}a_gb_hk$'
+description: 'The <a href="keywords/keyword/10">semigroup ring</a> $R[G]$ where $G$ is a group (not necessarily commutative).'
 name: group ring

db/ringapp/keyword/KWD_000022.yaml

@@ -1,2 +1,2 @@
-description: null
+description: The algebraic closure of a field $k$ is defined as an algebraic extension of $k$ that is algebraically closed. It is unique up to field isomorphism.
 name: algebraic closure

db/ringapp/keyword/KWD_000024.yaml

@@ -1,2 +1,2 @@
-description: null
+description: A ring $Q$ is usually called a right ring of quotients (or fractions) for a ring $R$ if there is a homomorphism $R \to Q$ that maps some multiplicative set $S \subset R$ to invertible elements of $Q$. There are many constructions of rings of quotients (applicable in different situations and generally inequivalent), including classical, maximal, and Martindale rings of quotients.
 name: ring of quotients

db/ringapp/keyword/KWD_000025.yaml

@@ -1,2 +1,2 @@
-description: null
+description: The construction uses a tensor product of rings over a base ring.
 name: tensor product

db/ringapp/keyword/KWD_000026.yaml

@@ -1,2 +1,2 @@
-description: null
+description: The free associative algebra over a commutative ring $R$ generated by a set $X$ of indeterminates (possibly infinite): the set of finite $R$-linear combinations of words in the elements of $X$.
 name: free algebra

db/ringapp/keyword/KWD_000027.yaml

@@ -1,3 +1,3 @@
-description: Given a ring $S$ and an endomorphism $\sigma:S\to S$, you take $S[x]$
-  and define $xa=\sigma(a)x$ instead of regular multiplication. (May be defined symmetrically.)
+description: Given a ring $S$ and an endomorphism $\sigma:S\to S$, take $S[x]$
+  and define $xa=\sigma(a)x$ instead of the regular polynomial multiplication. (The symmetrical definition need not give an isomorphic ring.)
 name: twisted (skew) polynomial ring

db/ringapp/keyword/KWD_000028.yaml

@@ -1,2 +1,2 @@
-description: Constructions using valuations
+description: Let $G$ be an ordered Abelian group. A valuation is a mapping $v$ from a ring to $G \cup \{ +\infty\}$ that satisfies the conditions $v(x) = \infty$ if and only if $x = 0$, $v(xy)$ = $v(x)+v(y)$, and $v(x-y) \ge \min(v(x), v(y))$.
 name: valuations

db/ringapp/keyword/KWD_000029.yaml

@@ -1,3 +1,2 @@
-description: Related to the construction with $i,j,k$ for quaternions and generalized
-  quaternions.
+description: Related to the construction of quaternions and generalized quaternions. A generalized quaternion algebra $\left(\dfrac{a,b}{F}\right)$ over a field $F$ of characteristic $\not = 2$ is defined as $F\langle i,j,k\rangle/(i^2-a, j^2-b, ij+ji, k-ij)$, where $a,b \in F \setminus \{0\}$. (In the classical construction, $a=b=-1$.)
 name: quaternion algebra

db/ringapp/keyword/KWD_000032.yaml

@@ -1,2 +1,2 @@
-description: Completion with respect to an ideal.
+description: "Let $I$ be an ideal of a ring $R$. The $I$-adic completion of $R$ is defined as the inverse limit of the rings $R/I^n$, that is, the subring of $\prod\limits_{i=0}^\infty R/I^n$ that consists of sequences $(a_i)$ such that $a_m \equiv a_k \mod I^k$ for all $m > k$."
 name: completion

db/ringapp/keyword/KWD_000033.yaml

@@ -1,3 +1,2 @@
-description: like a group ring but the coefficients don't necessarily commute with
-  the group elements.
+description: Let $k$ be a field, $G$ a group (possibly noncommutative). A twisted group ring $k^\gamma[G]$ is the set of finite formal $k$-linear combinations of elements of $k$ with pointwise addition and convolutional multiplication, but multiplying group elements introduces a scalar factor that depends on the elements: if $g_1 g_2 = g$ in $G$, then $g_1 g_2 = \gamma(g_1,g_2)g$ in $k^{\gamma}[G]$. The mapping $\gamma$: $G\times G \to k\setminus \{0\}$ is called a 2-cocycle; associativity forces it to fulfill the condition $\gamma(x,y) \gamma(xy,z) = \gamma(y,z) \gamma(x,yz)$.
 name: twisted group ring

db/ringapp/keyword/KWD_000034.yaml

@@ -1,3 +1,3 @@
 description: A procedure on commutative local rings that makes extends the ring to
-  a Henselian local ring.
+  a Henselian local ring. See <a href="https://stacks.math.columbia.edu/tag/0BSK">The Stacks project</a>.
 name: Henselization

db/ringapp/keyword/KWD_000035.yaml

@@ -1,2 +1,2 @@
-description: $R[M]$ where $M$ is a monoid.
+description: The <a href="keywords/keyword/10">semigroup ring</a> $R[M]$ where $M$ is a monoid.
 name: monoid ring

db/ringapp/keyword/KWD_000068.yaml

@@ -1,2 +1,8 @@
-description: Can be described as a Leavitt path algebra (https://en.wikipedia.org/wiki/Leavitt_path_algebra)
+description: "Let $K$ be a field and $G = (V,E)$ a directed graph. For each edge $e \in E$, denote by $s(e)$ its source, and by $r(e)$, its range. Consider the free associative algebra $F_G = K \langle V \sqcup E \sqcup E^* \rangle$ (the first two subsets of indeterminates correspond to the vertices and the edges, while the third consists of symbols $e^*$, where $e \in E$). The Leavitt path algebra $L_K(G)$ of $G$ over a field $K$ is defined as the quotient of $F_G$ by the following relations:\n\
+(V) $vv' = \delta_{vv'}v$,\n\
+(E1) $s(e)e=er(e)=e$,\n\
+(E2) $r(e)e^*=e^*s(e)=e^*$,\n\
+(CK1) $e^*e'=\delta_{ee'}r(e)$,\n\
+(CK2) $v = \sum\limits_{e \in E: s(e) = v} ee^*$.\n\
+The first four relations are imposed on all elements $v,v' \in V$ and $e, e' \in E$, while the last one is imposed only on such vertices $v$ that that the sum is finite and nonzero (such vertices are called regular)."
 name: Leavitt path algebra

db/ringapp/keyword/KWD_000071.yaml

@@ -1,5 +1,5 @@
 description: Let $K/F$ be a cyclic field extension of degree $n$, let $\sigma$ be
-  a generator of $Gal(K/F)$ and $u\in F^\times$.  Denote $(K/F,'\sigma, u)=\oplus_{i=0}^{n-1}Kz^i$
+  a generator of $Gal(K/F)$ and $u\in F^\times$.  Denote $(K/F,\sigma, u)=\oplus_{i=0}^{n-1}Kz^i$
   where $z$ is a symbol, and define a multiplication by $zx=\sigma(x)z$ and $z^n=u$.  This
   is a central simple $F$ algebra of degree $n$.
 name: cyclic algebra

db/ringapp/keyword/KWD_000072.yaml

@@ -1,4 +1,5 @@
-description: "Given a lattice ordered group $G$, In Jaffard, Paul. \"Les syst\xE8\
-  mes d'id\xE9aux.\" (1960), it is described how to find a Bezout domain with divisibility\
-  \ group $G$."
+description: "A partially ordered group is an Abelian group with a partial ordering $\le$ and $g\le h$ implies $g+i \le h+i$. A lattice ordered group is a partially ordered group such that binary infima and suprema always exist and are well-defined.\n\n\
+If $R$ is a commutative integral domain, $Q$ is its field of fractions, $Q^*$ is the multiplicative group of $Q$, and $U(R)$ is the group of units of $R$, the quotient group $Q^*/U(R)$ partially ordered by $aU \le bU$ whenever $a^{-1} b \in R$ is called the divisibility group of $R$.\n\n\
+A valuation on a field $Q$ by a totally ordered group $G$ is a map $v$: $Q^* \to G$ such that $v(x+y)\ge \inf(v(x),v(y))$ if $x+y \not = 0$, and $v(xy)=v(x)+v(y)$.\n\n\
+For any lattice ordered group $G$, there is a lattice embedding $f$: $G \hookrightarrow G' = \prod\limits_{M \in \Gamma} G_M$ into a direct product of totally ordered groups $G_M$ (with the product ordering); let $\pi_M: $G' \to G_M$ be the canonical projections. For the field $Q = k(\{Y_g: g \in G\})$, define a valuation $\phi_M$: $Q^* \to G_M$ as follows: for the monomials, $\phi_M(\c \prod\limits_{i=1}^r Y_{g_i}^{n_i}) = \sum\limits_{i=1}^r n_i \pi_M(f(g_i))$; for a polynomial, the valuation is defined to be the infimum of the valuations of its monomials; then naturally extend the map to quotients of polynomials. Now define $\prod\limits_{M \in \Gamma} \phi_M = \phi$: $Q^* \to G'$ and $R = \{0\} \cup \{x \in Q^*: \phi(x) \ge 0\}$. $R$ is the Jaffard-Ohm-Kaplansky construction of an integral domain with quotient field $Q$ and divisibility group $\phi(Q^*) = f(G) \cong G$."
 name: Jaffard-Ohm-Kaplansky construction

db/ringapp/keyword/KWD_000073.yaml

@@ -1,2 +1,4 @@
-description: ''
+description: "Let $p \in \Bbb Z$ be prime and $R$ a commutative ring. For $x = (x_0, x_1, \ldots) \in R^{\Bbb N}$, denote $x^{(n)} = \sum\limits_{i = 0}^n x_i^{p^{n-i}} p^i$. It is known that for $S = \Bbb Z[x_0, x_1, \ldots, y_0, y_1, \ldots]$ there exist sequences of polynomials $\alpha, \pi \in S^{\Bbb N}$ such that $\alpha^{(n)} = x^{(n)} + y^{(n)}$ and $\pi^{(n)} = x^{(n)} y^{(n)}$ for $x = (x_i)$, $y = (y_i)$ and every $n \ge 0$. The ring $W$ of Witt vectors is the Cartesian product $R^{\Bbb N}$, endowed with addition $a+b = (\alpha_0(a,b), \alpha_1(a,b), \ldots)$ and multiplication $ab = (\pi_0(a,b), \pi_1(a,b), \ldots)$, its unity is $(1,0,\ldots)$.\n\n\
+There are multiple variations of this construction, such as the large, $p$-typical, or truncated Witt vectors."
+
 name: ring of Witt vectors

db/ringapp/keyword/KWD_000075.yaml

@@ -1,3 +1,2 @@
-description: 'A construction of algebras using a bilinear form on a vector space:
-  https://en.wikipedia.org/wiki/Clifford_algebra'
+description: 'Let $k$ be a field, $V$ a vector space over $k$, $Q$ a quadratic form on $V$. The Clifford algebra $\mathrm{Cl}(V,Q)$ is defined as $T(V)/(\{v\otimes v - Q(v): v \in V\})$, where $T(V)$ is the tensor algebra of $V$.'
 name: Clifford algebra

db/ringapp/keyword/KWD_000076.yaml

@@ -1,2 +1,2 @@
-description: ''
+description: 'A Clifford algebra of a quadratic form that is identically zero.'
 name: Grassmann algebra

db/ringapp/keyword/KWD_000077.yaml

@@ -1,3 +1,3 @@
 description: Construction involves use of a direct limit of objects, for example,
-  a directed union.
+  a directed union of rings.
 name: direct limit

db/ringapp/keyword/KWD_000078.yaml

@@ -1,2 +1,2 @@
-description: ''
+description: "A basic ring of a <a href="keywords/keyword/25">semiperfect</a> ring $R$ is a corner ring $eRe$ for $e = e_1 + \ldots + e_r$, where $e_i$ are orthogonal primitive idempotents of $R$ such that $e_i R$ constitute a complete set of isomorphism classes of principal indecomposable right $R$-modules."
 name: basic ring