All Questions
1,966 questions with no upvoted or accepted answers
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Tensor product of preordered rings
All rings in this post are commutative, unital, and contain $\frac{1}{2}$.
To study "real" properties of a ring $R$, one is often interested in the orderings which exist on fraction fields of ...
- 277
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118
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Multiplicity of a polynomial in positive characteristic
Let $\mathbb K$ be a field of characteristic $p>0$.
Let $f\in\mathbb K[x_1,\dots,x_n]$ be a multivariate polynomial and let $q\in\mathbb K^n$. Is there a computational method to determine the ...
- 351
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153
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Locus of trivialization of an extension of a vector bundle
Let $X$ be a normal noetherian affine scheme. Let $j:U\rightarrow X$ be a codimension two open of $X$ and $\mathcal{E}$ a vector bundle on $U$.
We assume that $j_*\mathcal{E}$ is a vector bundle. In ...
- 3,472
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112
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Homogeneous basis on a polynomial subalgebra
Let $k$ be an algebraically closed field of characteristic $0$ and $A=k[X_1, \ldots, X_n]$ with the grading induced by the total degree. Let $B$ be a graded $k$-subalgebra of $A$, ie, if $(A_k)$ is ...
- 1
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287
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On the product in the power series ring
Let $A_n \colon= K[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $2n$ variables and ${\frak m}_{A_n}$ be the unique maximal ideal of $A_n$.
Suppose we have two ...
- 563
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129
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Homomorphisms from $k[x,y]$ to $k[x,x^{-1},y]$
Let $k$ be a field of characteristic zero and let $R_{-1}:=k[x,x^{-1},y]$
be the $k$-algebra of polynomials in $x,y$ containing the inverse of $x$, denoted by $x^{-1}$.
Let $f: k[x,y] \to R_{-1}$ be ...
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558
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Intersection of an ideal and a subring
Is there a Grobner basis method that can compute the intersection of an ideal $I$ of a polynomial ring $R$ and its subring $R^\prime$? For example, I have an ideal $I=(x+y+z^2,1+xyz+yz+xz)$ of $\...
- 245
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89
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Structures like vector spaces but closed under heterogeneous products
The category of (pseudo-)Euclidean vector spaces (vector spaces with a nondegenerate but not necessarily positive-definite quadratic form) is not closed under products because $R^n$ over $R$ and $Z_2^...
- 93
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141
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Two commuting matrices over a commutative ring
I would like to know if there are results about the dimension of the Algebra generated by two commuting Matrices over a ring (as there are in the case of a Field).
The good news is that "my" ring is ...
- 337
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126
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Reference request: J-P Serre, "Groupes finis d'automorphismes d'anneaux locaux réguliers"
Does anyone have, or know a link to, a copy of the paper named in the title? It is published in Colloq. d'Alg. École Norm. de Jeunes Filles, Paris (1967), 1-11. I do not have ready access to Serre's ...
- 3,137
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68
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Normality of certain subrings of polynomial rings in characteristic p
Let $k$ be an algebraically closed field of characteristic p. Let
$Z\subset k[x_1,\cdots,x_n]$ be a graded $k$-subalgebra of a
polynomial ring, such that for any $f\in Z,$ any divisor of $f$ (in
$k[...
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90
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Do Plucker relation follow from a subsystem of equations?
The following system of equation: $i, j \in \mathbb{Z}_{\ge 1}$, $i+1<j$, $J \subset \mathbb{Z}_{\ge 1}$, $J \cap \{i,j,i+1,j+1\} = \emptyset$,
\begin{align*}
P_{i,j,J}P_{i+1,j+1,J} = P_{i,j+1,J} ...
- 6,201
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448
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Behavior of Ext under base change
Let $x$ be a nonzerodivisor on a local Noetherian ring $(R,m).$ Let $M,N$ be finitely generated $R/xR$-modules.
How to show the existence of the following exact sequence
$\cdots\longrightarrow ...
- 1,713
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332
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Krull dimensions and regular sequences
I am trying to understand if a certain condition on quotient rings is sufficient for a sequence to be regular. Here is the setting:
Let $\mathbb{C}[u_1,...,u_n]$ be the ring of regular functions on $\...
- 1,227
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96
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How to write the involution in the new coordinates?
Let $f= xy^3+y^4-x^2+xy$. Using the following codes in Maple,
f := xy^3+y^4-x^2+xy;
v := Weierstrassform(f, x, y, x0, y0);
I obtain the following result:
\begin{align}
& f_0 = {{ x_0}}^{3}+{{...
- 6,201
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213
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make me idempotent
$T_n$ be the full transformation semigroup on $X_n= \{1, 2, \cdots , n\}$.
$D_r =\{\alpha \in T_n: |im(\alpha)|=r\}$.
$E(D_r)$ is the set of all idempotents of semigroup $T_n$.
$support(\alpha)=\{...
- 109
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106
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Kelly's theorem for quadratic polynomials
Let $f_1, \ldots, f_m$ be homogeneous irreducible quadratic polynomials in $\mathbb{C}[x_1, \ldots, x_n]$.
Assume that these polynomials are pairwise coprime.
Denote $P:= f_1 \cdot f_2 \ldots \...
- 2,576
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145
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If $F(x_1,0,\ldots,0)=(x_1,0,\ldots,0)$, then $F$ is bijective?
Let $k$ be an algebraically closed field of characteristic zero and let $f_1,\ldots,f_n \in k[x_1,\ldots,x_n]$ have an invertible Jacobian, namely, the determinant ot their Jacobian matrix belongs to $...
- 2,837
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136
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Amalgamated free-product of semigroups (definition)
I am self-studying some concepts including the title one. I reached the definition of an amalgamated free-product ${S_1}{*_U}S_2$ where $[S_1, S_2; U, w_1,w_2]$ is an amalgam of semigroups. Let $S_1=\...
- 233
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196
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Generalizations of 'Injectivity on one line'
The main result of J. Gwozdziewicz in this paper says the following:
"Let $k$ be an algebraically closed field of characteristic zero, and let $f:k[x,y] \to k[x,y]$, $(x,y) \mapsto (p,q)$, be a $k$-...
- 2,837
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53
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a generalization of group (monoid with order-by-order invertible elements)
Fix a filtered monoid, $H=H_0\supsetneq H_1\supsetneq H_2\supsetneq\cdots$. Suppose for any $h\in H$ and any $n\in \Bbb{N}$ exists $h_n\in H$ such that $h\cdot h_n\in H_n$ and $h_n\cdot h\in H_n$. If ...
- 2,232
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202
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finishing the proof of artin approximation
At the end of the proof of Artin's approximation theorem, and using all his notation, he reduces to finding a solution $y\in A$ such that
$$y\equiv\overline y\mod \mathfrak m^c$$
$$\tag{*}f(y)\equiv0\...
- 1,217
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242
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Quotient by augmentation ideal
Let $p$ be a prime number. Denote by $\mathbb{Z}_p$ the ring of $p$-adic integers and let $R = \mathbb{Z}_p [[X_1, \ldots, X_n]] / (f_1, \ldots, f_d)$.
Assume that a finite abelian group $G$ of order ...
- 183
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296
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Union of varieties
Let $Q_1, \ldots, Q_k$ and $P_1,\ldots, P_m$ be irredicable homogenous polynomials in $\mathbb{C}[x_0,\ldots, x_n]$ such that $V(Q_1, \ldots, Q_k) \subseteq \cup_i V(P_i)$. Here $V$ is projective ...
- 2,576
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100
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Solutions of the linear equation from K[[X_1,X_2,X_3]] to K[[X_1,X_2]]
Let $A_3 := K[[X_1,X_2,X_3]]$ be a three-variable formal power series ring over a field $K$. We consider a linear equation
$(\sharp) \phantom{aa} a_1(X_1,X_2,X_3)Y_1 + \ldots + a_n(X_1,X_2,X_3)Y_n = ...
- 563
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303
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For which monic irreducible $f \in \mathbb{C}[x,y][T]$, $\mathbb{C}[x,y][T]/(f)$ is a UFD?
Let $f=f(T) \in \mathbb{C}[x,y][T]$ be a monic irreducible polynomial:
$f=T^n+a_{n-1}T^{n-1}+\cdots+a_1T+a_0$,
$a_j \in \mathbb{C}[x,y]$, $0 \leq j \leq n-1$.
Denote $B=\mathbb{C}[x,y,T]/(f)=\mathbb{C}...
- 2,837
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213
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For which $f \in \mathbb{C}[x,y][T]$, all irreducible elements of $\mathbb{C}[x,y]$ remain irreducible in $\mathbb{C}[x,y,T]/(f)$
Let $f=f(T) \in \mathbb{C}[x,y][T]$ be a monic polynomial:
$f=T^n+a_{n-1}T^{n-1}+\cdots+a_1T+a_0$,
$a_j \in \mathbb{C}[x,y]$.
Denote: $A=\mathbb{C}[x,y]$ and $B=\mathbb{C}[x,y,T]/(f)=\mathbb{C}[x,y][...
- 2,837
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309
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Proof of Krull's intersection theorem with Taylor expansion
I asked this question last year in MSE, but I didn't get an answer.
I took a commutative algebra course last semester (using Kaplansky's book), and I learned about Krull's intersection theorem. In ...
- 2,215
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101
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Spherical Rings
My question is concerned with filtered rings. It is a classical result that if $R$ is a finitely generated commutative ring graded by a semigroup $S$ then $S$ is also finitely generated.
The reverse ...
- 501
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81
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A quaternion x generates a left ideal of rank 2 if and only if x, ix and jx are linearly dependent?
I am trying to understand the construction of Artin and Mumford of a non-rational unirational threefold in ([1], p.90).
Assume $S$ is a smooth projective surface over $\mathbb{C}$ with a smooth ...
- 1,025
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75
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Homomorphic image of $B_{\lambda}^o(S)$ is the Brandt $\lambda^o$-extension of some monoid with zero
Let $S$ be a monoid with zero and $I_{\lambda}$ be an indexed set, then $B_{\lambda}(S) = \{ (\alpha, s , \beta ) : \alpha , \beta \in I_{\lambda}, s\in S \} \cup \{0\}$ is a semigroup and $J = \{ (\...
- 143
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94
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Is the tensor product of two commutative semiprime Q-algebras semiprime?
A ring is semiprime if it has no non-zero nilpotents. Let $Q$ denote the rational numbers and $A,B$ be a pair of commutative semiprime $Q$ algebras. Is $A\otimes_Q B$ semiprime? It is well known that ...
- 383
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251
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"Strong Going-Down" Theorem
Let $\iota \colon A \subset B$ be a finite integral extension between domains. Suppose that $A$ is UFD, so $A$ is an integrally closed domain.
$A$ and $B$ may not be noetherian ring.
Choose a prime ...
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112
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$F$-pure threshold of an $F$-pure ideal
According to this reference an elliptic curve is $F$-pure if and only if the $F$-pure threshold of its defining ideal is $1$. Does there exist an $F$-pure local ring $R=A/\mathfrak{a}$ such that $\...
- 591
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289
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Quotient of Cohen-Macaulay ring
Let $A$ be a Cohen-Macaulay ring, and $I$ an ideal of $A$.
What can we say about $\operatorname{depth}(A/I)$?
I know that $\operatorname{depth}(A/I)\le \dim(A/I)$.
- 359
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138
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Properties of a subring of a 'completion' of k(X_1, X_2, ..., X_n)
I'm looking for a reference in commutative algebra for the properties of the ring made of polynomials in $n$ indeterminate over a field $k$ with "real exponents".
I don't even know the name of this ...
- 101
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168
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In commutativity theorems in ring theory
Suppose that $R$ is a ring such that for any $x\in R$ there exists $1<n(x)\in \mathbb{N}$ such that $x^{n(x)}-x\in Z(R)$. Prove that $R$ is commutative or if it is not commutative, then the ideal ...
- 221
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125
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Embedding a cancellative monoid into another in such a way that $|X-x|=|X|$, where $X$ is a fixed finite set and $x\in X$
Preliminaries.
Let $\mathbb A = (A, +)$ be a possibly non-commutative semigroup. For $X, Y \subseteq A$ we set
$$
X - Y := \{a \in A: a + y \in X\text{ for some }y \in Y\},
$$
which is just the usual ...
- 10.5k
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165
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on the ``generic" modules of finite length (skyscrapers)
Let $R$ be a local or graded ring. (If it helps, can assume the ring is "good", e.g. $R=k[[x_1,..,x_p]]$, where $k$ is a field of zero characteristic.)
Let $M$ be a finitely generated $R$-module ...
- 2,232
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82
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grade of ideals in non-noetherian rings
Let $R$ be a commutative ring with unity, and $M$ an $R$-module. Assume that $I$ and $J$ are finitely generated ideals and $K$ another ideal of $R$. Let $\textbf{x}$ be a sequence of generators of $I$...
- 1,355
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0
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109
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Non-local differentially smooth algebra
Let $A$ be a noetherian commutative algebra over a perfect field $k$.
The algebra $A$ is said to be differentially smooth over $k$ if
(1) $\Omega^1_{A/k}$ is a projective $A$-module, and
(2) the ...
- 657
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308
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Basic question about power series and complete group algebras
This is a pretty basic question, but I suspect it might be too exotic for math.stackexchange.
Let $\mathbb{Z}_p$ be the $p$-adic integers. For free pro-$p$ group $F_r$ of rank $r$, we can consider ...
- 10.7k
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197
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Name of some commutative ring akin to $p$-adics
I need help in identifying the naming convention of some commutative ring described below.
Let $p$ be a prime, let $k$ be a positive integer, and let $$\textbf{e} = (e_0,\ldots,e_{k-1})$$ be a list ...
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301
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Irreducible component of a scheme over a dvr
Let $\mathcal M$ be a (reduced) quasi-projective scheme over a dvr (of mixed caracteristics), $R$. Suppose that the generic fiber $\mathcal M_{\eta_R}$ is (nonempty) smooth and irreducible of ...
- 155
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282
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Proving the algebraic independence of certain elements
Let $k$ be a field of characteristic zero and $R$ be the polynomial ring $k[x_1,...,x_n,t_1,...,t_n]$. Let $P_i = (a_{i1}:a_{i2}:a_{i3})$ be $n$ points in the projective plane over $k$, such that not ...
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63
views
Writing a module as a direct sum
Let $q_1, q_2, q_3 \in \mathbb{Z}[x,y]$ such that $q_1, q_2$ are algebraically independent and let $S$ be an algebra generated by $q_1, q_2, q_3$ over $F_p$. If writing $S$ as a module over $F_p [...
- 11
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177
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Intersections of ideals and nilpotence
Let $R$ be a polynomial ring over a field $k$, $R = k[x_1, \dots, x_n]$. Suppose $R'$ is an associative $R$-algebra and it has the property that there exists a degree $m<n$ monomial in the $x_i$'s ...
- 25.6k
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112
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A question about a specific inverse proposition of Combinatorial Nullstellensatz
From the Hilbert's Nullstellensatz, we have the following consequence which is usually called Combinatorial Nullstellensatz:
Let $F$ be an arbitrary field, and let $f = f(x_1,x_2,\cdots,x_n)$ be a ...
- 1,997
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151
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book for help on problems with noetherian rings
Can you please introduce to me a book which would help me to prove the two following problems?
In a noetherian ring, every integrally closed ideal is unmixed.
Let $R$ be a noetherian ring, $P$ a ...
- 565
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176
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Flatness of a simple ring extension
Assume $A \subseteq B=A[b]$ are integral domains, $b \in B$ is algebraic over $A$ (but not necessarily integral over $A$), and $A$ and $B$ have the same field of fractions.
(Notice that $b=u/v$ for ...
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