diff --git a/db/ringapp/keyword/KWD_000001.yaml b/db/ringapp/keyword/KWD_000001.yaml index d9bc9627..8ac39899 100644 --- a/db/ringapp/keyword/KWD_000001.yaml +++ b/db/ringapp/keyword/KWD_000001.yaml @@ -1,2 +1,2 @@ -description: null +description: The ring of fractions with numerators and denominators from a commutative polynomial ring (possibly multivariate). Often called the ring of rational functions. name: rational polynomial ring diff --git a/db/ringapp/keyword/KWD_000002.yaml b/db/ringapp/keyword/KWD_000002.yaml index 625b1412..1997ef71 100644 --- a/db/ringapp/keyword/KWD_000002.yaml +++ b/db/ringapp/keyword/KWD_000002.yaml @@ -1,2 +1,2 @@ -description: null +description: The polynomial ring $R[x]$ is the monoid ring 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 diff --git a/db/ringapp/keyword/KWD_000003.yaml b/db/ringapp/keyword/KWD_000003.yaml index 9c7a31f4..c384390e 100644 --- a/db/ringapp/keyword/KWD_000003.yaml +++ b/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 rings of quotients. name: quotient ring diff --git a/db/ringapp/keyword/KWD_000005.yaml b/db/ringapp/keyword/KWD_000005.yaml index 4a79c2d1..4da264d1 100644 --- a/db/ringapp/keyword/KWD_000005.yaml +++ b/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 rings of quotients. name: localization diff --git a/db/ringapp/keyword/KWD_000008.yaml b/db/ringapp/keyword/KWD_000008.yaml index 1d679354..54f43d43 100644 --- a/db/ringapp/keyword/KWD_000008.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000009.yaml b/db/ringapp/keyword/KWD_000009.yaml index 5f4e26b8..f030209a 100644 --- a/db/ringapp/keyword/KWD_000009.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000010.yaml b/db/ringapp/keyword/KWD_000010.yaml index 144f82bd..ebf87091 100644 --- a/db/ringapp/keyword/KWD_000010.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000013.yaml b/db/ringapp/keyword/KWD_000013.yaml index 1311c3fc..d54c3737 100644 --- a/db/ringapp/keyword/KWD_000013.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000014.yaml b/db/ringapp/keyword/KWD_000014.yaml index f222efbb..d0420eed 100644 --- a/db/ringapp/keyword/KWD_000014.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000018.yaml b/db/ringapp/keyword/KWD_000018.yaml index 94a4d2bb..e2dc26c7 100644 --- a/db/ringapp/keyword/KWD_000018.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000019.yaml b/db/ringapp/keyword/KWD_000019.yaml index 7dfdc6aa..43e785ce 100644 --- a/db/ringapp/keyword/KWD_000019.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000021.yaml b/db/ringapp/keyword/KWD_000021.yaml index 866c60a4..817f948a 100644 --- a/db/ringapp/keyword/KWD_000021.yaml +++ b/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 semigroup ring $R[G]$ where $G$ is a group (not necessarily commutative).' name: group ring diff --git a/db/ringapp/keyword/KWD_000022.yaml b/db/ringapp/keyword/KWD_000022.yaml index 9246f8de..3215cb50 100644 --- a/db/ringapp/keyword/KWD_000022.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000024.yaml b/db/ringapp/keyword/KWD_000024.yaml index a9435f18..b0991a40 100644 --- a/db/ringapp/keyword/KWD_000024.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000025.yaml b/db/ringapp/keyword/KWD_000025.yaml index 5fbb0500..a7bba65c 100644 --- a/db/ringapp/keyword/KWD_000025.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000026.yaml b/db/ringapp/keyword/KWD_000026.yaml index be82299b..172f10f3 100644 --- a/db/ringapp/keyword/KWD_000026.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000027.yaml b/db/ringapp/keyword/KWD_000027.yaml index 47efcfdf..3c68bc22 100644 --- a/db/ringapp/keyword/KWD_000027.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000028.yaml b/db/ringapp/keyword/KWD_000028.yaml index 77dd5bf8..db39255b 100644 --- a/db/ringapp/keyword/KWD_000028.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000029.yaml b/db/ringapp/keyword/KWD_000029.yaml index 1cbdee9a..97b7bb2e 100644 --- a/db/ringapp/keyword/KWD_000029.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000032.yaml b/db/ringapp/keyword/KWD_000032.yaml index 5159af5d..b88eb135 100644 --- a/db/ringapp/keyword/KWD_000032.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000033.yaml b/db/ringapp/keyword/KWD_000033.yaml index 184a2beb..cc349e6d 100644 --- a/db/ringapp/keyword/KWD_000033.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000034.yaml b/db/ringapp/keyword/KWD_000034.yaml index a9c2d853..1ba11c0b 100644 --- a/db/ringapp/keyword/KWD_000034.yaml +++ b/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 The Stacks project. name: Henselization diff --git a/db/ringapp/keyword/KWD_000035.yaml b/db/ringapp/keyword/KWD_000035.yaml index a7dbcbc2..e68f5c14 100644 --- a/db/ringapp/keyword/KWD_000035.yaml +++ b/db/ringapp/keyword/KWD_000035.yaml @@ -1,2 +1,2 @@ -description: $R[M]$ where $M$ is a monoid. +description: The semigroup ring $R[M]$ where $M$ is a monoid. name: monoid ring diff --git a/db/ringapp/keyword/KWD_000068.yaml b/db/ringapp/keyword/KWD_000068.yaml index 02f7d762..8f4a023e 100644 --- a/db/ringapp/keyword/KWD_000068.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000071.yaml b/db/ringapp/keyword/KWD_000071.yaml index ade53cfc..b1bf11aa 100644 --- a/db/ringapp/keyword/KWD_000071.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000072.yaml b/db/ringapp/keyword/KWD_000072.yaml index 7471f28a..54772c05 100644 --- a/db/ringapp/keyword/KWD_000072.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000073.yaml b/db/ringapp/keyword/KWD_000073.yaml index c6f2ea18..9ad0e041 100644 --- a/db/ringapp/keyword/KWD_000073.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000075.yaml b/db/ringapp/keyword/KWD_000075.yaml index b2b10add..f5e69bd6 100644 --- a/db/ringapp/keyword/KWD_000075.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000076.yaml b/db/ringapp/keyword/KWD_000076.yaml index aa1397ed..04dd76f8 100644 --- a/db/ringapp/keyword/KWD_000076.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000077.yaml b/db/ringapp/keyword/KWD_000077.yaml index 1e434a1d..3b0ab67b 100644 --- a/db/ringapp/keyword/KWD_000077.yaml +++ b/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 diff --git a/db/ringapp/keyword/KWD_000078.yaml b/db/ringapp/keyword/KWD_000078.yaml index 8a317f64..4f148654 100644 --- a/db/ringapp/keyword/KWD_000078.yaml +++ b/db/ringapp/keyword/KWD_000078.yaml @@ -1,2 +1,2 @@ -description: '' +description: "A basic ring of a semiperfect 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