# Tagged Questions

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### If $B\subseteq A$ are free & finite rank $R$-algebras, is $R\to A \otimes_B R$ injective?

(In this question, all rings and algebras are commutative with identity.) I have a situation that boils down to the following data: a ring $R$, an $R$-algebra $A$ with a subalgebra $B$ such that $A$ ...
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### Vertices of a polytope as algebra generators

I am wondering if the following kind of objects has some name, or are there any studied examples. I apologize for perhaps too specific definition, this is an adoptation of a situation that arises in ...
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### Injective flat module

Let $R$ be a (right noetherian) ring. Is there always a right $R$-module which is both flat and injective? If $R$ is an integral domain, then the answer is indeed yes, as the quotient field is such. ...
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### Global dimension of a subalgebra with all units

(All rings here are always assumed to be unital and associative). Setup Let $R$ be a ring, and $A$ and $B$ be $R$-algebras, with $A$ a commutative subalgebra of $B$ satisfying: If $u$ is a unit ...
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This question was asked earlier on math.stackexchange: click here. See the comments and the answer by Jack Schmidt there. Let $M$ be a module over a commutative ring $R$. It is possible that $M ... 2answers 166 views ### Is there an intuitionistic generalized boolean algebra (of Stone)? A "boolean algebra without the greatest element" was called by Stone "generalized boolean algebra" and he axiomatized it. Is there any publication about "preudo-boolean algebras without the greatest ... 1answer 214 views ### Inverse limit of Gorenstein local rings is again Gorenstein? If we have the system of surjective ring homomorphisms$f_{i,i+1}: R_{i+1} \twoheadrightarrow R_i$for an arbitrary$i \geq 0$such that all$R_i$are Gorenstein local ring. Let us put ... 0answers 84 views ### Super-Gorenstein ideal of${\Bbb F}_p[[X_1,\ldots,X_n]]$Let$A \colon= {\Bbb F}_p[[X_1,\ldots,X_n]]$be a$n$-variable power series ring over a finite field${\Bbb F}_p$. We put${\frak m}_A \colon= (X_1,\ldots,X_n)$. Definition(Super-Gorenstein ideal): ... 2answers 248 views ### Rank of a$ \mathbb{Z}_{p}[[T]] $module Let$p$be a prime and$M$is a finitely generated$ \mathbb{Z}_{p}[[T]] $module. Suppose$M[p]$denotes the$p$-torsion of$M$. Then$M[p]$and$M/(p)$are both$ F_{p}$vector spaces. So we can ... 0answers 97 views ### Universally catenary and all its formal fibers over minimal members are Cohen-Macaulay but it has a nonCohen-Macaulay formal fiber Please help me to find a Noetherian local ring$R$such that:$R$is universally catenary and all its formal fibers over minimal members of$Spec(R)$are Cohen-Macaulay but$R$has a nonCohen-Macaulay ... 0answers 159 views ### Square of primary ideals Is there any example of a$P$-primary ideal$I$in a noetherian domain$R$such that$I^2=PI \not=P^2$? 1answer 184 views ### Algebra structure$Tor(A,A)$This is a question i asked on math.stackexchange but i didn't get any answer. Let$A$be algebra over commutative ring$k$and$P_{\bullet}=(P_i,d_i)\rightarrow A$,$k$projective resolution. Then we ... 1answer 116 views ### Localizations of hereditary rings It is known that if a commutative Noetherian ring$R$is hereditary then for any maximal ideal$M$the localization$R_M$is also hereditary. Is the Noetherian assumption necessary? 1answer 147 views ### Depth of polynomial ring$S=\Bbb{R}[x_1,x_2,x_3,…,x_n,…]$Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e,$S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. My Question is about the depth of this ring. Question: Could we ... 1answer 183 views ### chain of prime ideals in polynomial ring$S=\Bbb{R}[x_1,x_2,…,x_n,…]$Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e,$S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. Is the following question true? Question: Is there any maximal ... 1answer 227 views ### Bernstein-Sato polynomial (one variable) Let$R = \mathbb{C}[x_1,...,x_n]$,$p \in R$. There exists a monic (of lowest degree)$b_p(x) \in \mathbb{C}[s]$and a differential operator$D(s)$such that $$b_p(s) p^s = D(x)p^{s+1}.$$ The ... 2answers 184 views ### Tychonoff spaces and ideals Let$X$be a tychonoff space and let$T$be the set of all$f \in C(X)$such that for any$g$the equation$fg = 1$has at most finitely many solutions. Under what conditions on$X$, the set$T$is an ... 0answers 156 views ### Comparing different Euclidean algorithms on a Euclidean domain I have posted this question at stackexchange (502413), without responses until now. In the papers by T. Motzkin: The Euclidean Algorithm, Bull. AMS 55, 1949, pp. 1142--1146 and P. Samuel: About ... 3answers 310 views ### A question concerning the isomorphic type of continuous functions let$S$be the ring of all functions$f : \Bbb{R}\longrightarrow \Bbb{R}$which are continuous outside a bounded open interval containing zero (depended on$f$). Is it possible to consider$S$as ... 1answer 421 views ### Uncountable Reduced ring$R$with$R[x]$has only a countable number of maximal left ideals The question is following: Is there an uncountable reduced ring (i.e., a ring with no non-zero nilpotent element)$R$(with identity) such that$R[x]$has only a countable number of maximal ... 2answers 291 views ### Is there a Tychonoff space$X$of cardinality not of the form$2^\alpha$such that$|C(X)| = |X|$Let$X$be the real line with the usual topology. Then clearly$|C(X)| = c = |X|$and on the other hand$|X| = 2^{\aleph_0}$. Now my question is as in the title: Is there a Tychonoff space$X$of ... 2answers 351 views ### Is it true that if$M$is injective then$S^{-1}M$is also injective Let$R$be a commutative ring with identity and let$S$be a multiplicative subset of$R$. Is it true that for any injective$R$-module like$M$,$S^{-1}M$(as the$S^{-1}R$-module) is also injective ... 1answer 165 views ### elementary question on a completion of a ring Let$k$a field, and$k[\epsilon]=k[X]/(X^{2})$, what is the completion of the ring$k[\epsilon][t]$with respect to the ideal$(t^{2}+\epsilon)$? 1answer 207 views ### Symmetric algebra of an ideal and syzygies Let$(R,\mathfrak{m})$be a Cohen Macaulay local ring and$I=(a_1,\ldots,a_g,a_{g+1})$an almost complete intersection ideal of codimension$g.$Let$R^k\longrightarrow R^{g+1}\longrightarrow ...
Let $A$ be an Azumaya algebra over the commutative ring $R$ and let $\newcommand{\tr}{\operatorname{tr}} \tr\colon A \to R$ be the reduced trace as defined (for example) in Section IV.2 of M.-A. Knus, ...