All Questions
6,056 questions
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511
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Algebraic independence in polynomial rings over $\mathbb{Z}_n$
Let $R<S$ be an extension of commutative rings with identities (i.e. $R$ is a subring of $S$). We say $s_1,s_2,...,s_k$ are algebraically independent over $R$ iff there is no polynomial $f\in R[...
- 1,117
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2
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168
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Rank of a symmetric ideal
Let $\Sigma_m$ be the permutation group on $m$ letters and $R=\mathbb{C}[x_1,x_2,\dots,x_m]$. Let $\Sigma_m$ act on $R$ in the usual way. Let $R^{\Sigma_m}$ denote the ring of $\Sigma_m$ invariant ...
- 422
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2
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471
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Prime ideal of $A[X_1,...,X_d]$
Let $A$ be a UFD domain, i.e. integral and for any height one prime
${\frak p}$ of $A$, we have ${\frak p} = (u_{\frak p})$ for some $u_{\frak p} \in A$.
Once and for all, we fix the algebraic ...
- 781
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1
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500
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A question on Eisenbud-Green-Harris conjecture
Let $I$ be a homogeneous ideal in a polynomial ring $k[x_1,\ldots,x_n],$ $I$ contain a regular sequence $f_1,\ldots,f_n$ such that deg$(f_i)=a_i$ and $a_1\leq\ldots\leq a_n.$ Let $d$ be a non-negative ...
- 1,713
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1
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258
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How to compute graph ideal or cut ideal of a graph?
Graph ideals are a special case of Stanley-Reisner ideal, explained in Combinatorial Commutative Algebra book by Sturmfels, and graph ideals here. Graph ideals are generated by the minimal paths while ...
- 143
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1
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281
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Module-finiteness over the fixed subring
SETUP: Let $R$ be a finitely generated, noetherian, integral domain of characteristic $0$. Let $G$ be a finite group that acts on $R$ by ring automorphisms.
QUESTION: Is $R$ a finitely generated ...
- 687
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1
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759
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Does every prime power generate a primary ideal?
Let $R$ be a commutative ring with identity and let $p\in R$ be a prime element (i. e. $(p)$ is a prime ideal). If $R$ is an integral domain, it can be shown that $(p^k)$ is a primary ideal for every $...
- 119
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1
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214
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Hilbert function and numerical polynomial
Let $R=\bigoplus_{ n\in\mathbb N}R_{ n}$ be a Noetherian standard ring defined over an Artinian local ring. Let $M=\bigoplus_{ n\in\mathbb N}M_{ n}$ be an $\mathbb N$-graded $R$-module (not ...
- 1,713
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139
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What can be said about $A$ and $B$ given the exact sequence $0 \to R^p \to A \to R^r \to R^q \to B \to 0$?
Let $A,B$ be two $R$-modules over a commutative ring $R$ (restrict to $R = \mathbb{Z}$ or $R= \mathbb{K}$ a field where appropriate). Suppose $A$ and $B$ fit into an exact sequence
$0 \to R^p \to ...
- 11
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1
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134
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Theorem 16 , Chapter 5 of Northcott 's, Finite Free Resolutions: p.grade
Let $R$ be a commutative ring with identity. D.G. Northcott's, Finite Free Resolutions, has:
and in Theorem 16 of Chapter 5 proves that:
$p.grade(I,M) = p.grade(P,M)$ for some prime ideal $P$ ...
- 1,355
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1
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466
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Bertini-type theorem in positive characteristic [closed]
Let $f:X \to Y$ be a morphism of finite type of irreducible schemes over an algebraically closed field of characteristic $0$. Assume that $Y$ is non-singular. Let $x \in X$ be a closed point and $T_xf:...
- 503
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1
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255
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Socle of Almost Complete Intersections
Let $(A,m)$ be a complete Artinian local ring over a field $K$.
We focus on almost complete intersection ring $A$ of the form
$A = K[[X_1,...,X_N]]/(f_1,...,f_{N+1})$.
We assume that none of $f_i$ ...
- 781
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1
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224
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M is an R-module which is not finitely generated. is it true that $\inf \{ i| H^i_I(M)\neq 0 \}\le ht_M I?$
Let $R$ be a commutative noetherian ring (with $1$), $I$ an ideal of $R$, and $M$, a finitely generated $R$-module such that $IM \neq M$. Then, by Theorem 6.2.7 of BRODMANN-SHARP's Local Cohomology ...
- 1,355
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1
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383
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Dimension of Ext modules [closed]
Let $(R,m)$ be a noetherian local ring, and $M$ and $N$ be two finitely generated $R$-module. Then is it true that $\dim \text{Ext}^k(M,N)\leq \dim M-k$? If not does the reversed inequality hold?
- 21
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143
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Bound on the weight of the minimum weight generator of [n,k] cyclic codes?
I'm looking at creating sparse generator matrices for cyclic codes of a given length and dimension. A generator matrix of an [n,k] cyclic code can be expressed as
$G = \begin{bmatrix}g_0 & g_1 &...
- 111
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1
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845
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Relation between intersection and product of ideals
Let $C$ be a smooth projective (irreducible) curve in $\mathbb{P}^n$ for some $n$. Denote by $I_C$ the ideal of $C$. Let $g \in I_C\backslash I_{C}^2$, an irreducible element. Is it true that for any ...
- 2,032
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196
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Lax monoids where only the unit triangle is lax
I was rereading the paper Directoids: algebraic models of up-directed sets by Ježek and Quackenbush, this time with category theory in mind. When I tried to describe what the results in that paper ...
- 327
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1
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208
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betti-numbers of Gin(I), generic initial ideal of $I$
here in the paper Ideals with Stable Betti Numbers there is a theorem that I can't uderstand it, both in details (which highlighted) and sketch of the proof of (b):
can you help please?
...
- 1,355
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362
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Bounded dg algebra vs unbounded dg algebras
1)Let $Cd_{\geq 0}ga$ be the category of non negatively commutative cochain dg algebra over a field $\Bbbk$ of charachteristic zero. Let $w\: : \: Cd_{\geq 0}ga\to dg_{\geq 0}Mod$ be the forgethfull ...
- 1,424
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1
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272
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Self-similarity for simple algebraic structures [closed]
I'm doing this thread because I have some ideas about how to define self-similarity in algebra, but I don't know if this is known at all. Any critics, comments and references are more than welcomed. ...
- 438
1
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1
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323
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F-splitting and F-purity from commutative algebra viewpoint
First I define two terms:
Let $R$ be a commutative ring with identity,let char$R$ = $p$, let $F:R\rightarrow R$ be the Frobenius ring homomorphism. This makes $R$ into an $R$-module with respect to ...
- 11
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1
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361
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Algorithm for Polynomial Reduction in a Quotient Ring
Any reference or suggestion for the following problem would be greatly appreciated.
I am working on the quotient ring $Q=R[X_1,\dots,X_n]/<f_1,\dots,f_k>$. Given polynomials $p$ and $q$ I want ...
- 1,256
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218
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Does projective duality preserve arithmetic-Cohen-Macaulay-ness?
Let $V$ be a vector space over $\mathbb{C}$.
Suppose $X\subset \mathbb{P} V$ is an algebraic variety, and consider its projective dual variety $X^\vee \subset \mathbb{P} V^*$. If the coordinate ring $\...
- 569
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167
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Is it possible for a MCM module over a hypersurface to have infinite injective dimension?
Let $(R,\frak{m})$ be a hypersurface (i.e., $R=Q/(f)$, where $Q$ is a regular local ring, $0\not=f\in Q$). If $M$ is a MCM $R$-module, is it possible for the injective dimension of $M$ over $R$ to be ...
- 13
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159
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Relation between local cohomology and koszul cohomology of multigraded ring
Let $R$ be a ${\mathbb {Z}}^k$-graded Noetherian ring, $J=(x,y)$ an ideal of $R$ where $x,y\in R_{(1,\ldots,1)}.$ Is this following true $$H_{J}^i(R)=\underset{n}\varinjlim{H^i((x^n,y^n),R)},$$ where $...
- 127
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688
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Direct image of an ideal sheaf along a blow-up
Suppose that $I\subseteq\mathbb{C}[x_0,\ldots,x_n]$ is a saturated homogeneous ideal. Let $\mathcal{I}\subseteq\mathcal{O}_{\mathbb{P}^n}$ denote the corresponding coherent ideal sheaf, and then let $$...
user25585
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114
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Decomposition of skew-symmetric maps
Let $A$ be a ring and let $F$ be a finitely generated, free $A$-module. Let $\alpha: F \to \textrm{Hom}_A (F, A)$ be a skew-symmetric homomorphism, i.e. $\alpha(x)(y)=-\alpha(y)(x)$ for all $x,y \in F$...
- 3,031
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2
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375
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When is a power of an indeterminate in an ideal with 2 generators?
If I have an ideal ${\frak a} \colon= (f(X,S), g(X,S))$ of height $2$ in ${\Bbb C}[X, S]$, is it easy to know what power of $S$ is contained in $\frak a$? For example, what is the minimal number $m$? ...
- 87
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202
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Obstruction map for local singularities via tangent (Andre-Quillen) cohomology
Let $R$ be a local singularity (for example $R=\mathbb{C}[[x_1, \ldots , x_n]]/I$) ring over $\mathbb{C}$. Let $\mathbb{L}_{R}$ be a cotangent complex of $R$, then one can define tangent (Andre-...
- 1,545
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172
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Does this solution guarantee $det(A)=0$ where $A\in M(R)$? [closed]
Suppose $R$ is a commutative ring with identity $1$ and the following matrix equation holds:
$\begin{pmatrix} a_n & & \\
\vdots & \ddots & \\
a_1 &...
- 398
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1
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211
views
Is there a prime of height $i$ in support of $H^i_I(R)$?
$I$ is an ideal of a local Noetherian ring $R$ and $i>0$ .
Clearly the height of primes in support of $H^i_I(R)$ is at least $i$
The question is if it
contains a prime of height $i$, specially ...
- 189
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1
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396
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Embedded associated prime
$\underline{\textbf{Embedded associated prime}}$
I am reading the book "Joins and Intersections". In the proof of Rees theorem I have some doubt.
Let $\mathbf M$ be a finitely generated $\mathbf A$-...
- 31
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72
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Transformation terminology question
Given a transformation $t$ from the transformation semigroup $T_{n}$, if you take powers of $t$ under composition you get a length $s$ stem followed by a cycle. Permutations by definition have a ...
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1
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193
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Union of Associated Primes.
Let $R$ be a Noetherian ring. Let $I=(x_1,x_2,...,x_t)$ be a nonzero ideal of $R$. Define $I_n=(x_1^n,x_2^n,...,x_t^n)$. Are there any results about finiteness of $\cup_n Ass_R(I^n/I_n)$?
More ...
- 11
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534
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Initial ideal of k-th power of an ideal
Hi,
Let $I$ be an ideal in a polynomial ring $S = k[x_1, \ldots, x_n]$, where $k$ is an algebraically closed field of characteristic zero. Fix a term order on
$S$ (e.g. a lexicographic order) and ...
- 893
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1
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248
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Poset axioms of Boolean algebra [closed]
I found in Awodey's "Theory Category", second edition p34, a set of poset axioms to define Boolean algebras, whereas I don't see how they can be sufficient.
Here are the axioms:
A Boolean algebra ...
- 517
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2
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337
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Transformation of a bivariate polynomial into a homogeneous one
For a given a bivariate polynomial $P(x,y)$ with rational coefficients:
Q1. How compute such (invertible) substitutions of its variables that would transform the polynomial into a homogeneous one? In ...
- 34.3k
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127
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Degree bound for power of ideal
Let $I$ be an ideal in a commutative graded ring $R$, $M$ be a finitely generated graded $R$-module. Let $\varepsilon(M)$ be the smallest degree of a homogeneous element of $M$. An ideal $J$ is called ...
- 87
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1
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216
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Counting modular squares in an interval
For an integer $m$, let $S^m_{x_0,x_1} = \{ t | x_0 ≤ t ≤ x_1 $ and $t$ is a square modulo $m \}$. Let $S^m_x$ = $S^m_{0,x}$.
Determining whether the sets $S^m_x$ are empty is easy (1 is always a ...
- 123
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178
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Surjective and injective criteria via Hilbert polynomials
Let $X$ be a projective complex manifold. Is it true that any coherent sheaf $\mathcal{E}$ on $X$ whose Hilbert polynomial is a constant has a surjective morphsim $\mathcal{O}_X\rightarrow \mathcal{M}$...
- 13
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1
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230
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On the paper "On the asymptotic linearity of Castelnuovo-Mumford regularity"
I have posted this question on MSE, however it seems not to be interested by member there, so I decided to post it here. I am sorry if you feel it is not appropriate for MO.
I am now reading the ...
- 325
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1
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438
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Filter-regular sequence and regularity
Let $A$ be a commutative Noetherian ring, $R$ be a standard graded algebra over $A$, $M$ be finitely generated graded $R$-module. Let $R_{+}$ be the irrelevant ideal. The Castelnuovo-Mumford ...
- 325
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222
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Special case of testing integer polynomials for irreducibility
How much easier is testing polynomials of the form x^n + ax + b for irreducibility (in Z[x]) than testing polynomials in general? I am especially interested in the case where n is prime, which may be ...
- 11
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1
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334
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Number of generators of $\mathfrak m$-primary ideals in $k[x, y]$
Let $R = k[x, y]$ with $k$ algebraically closed, and $\mathfrak m = (x, y)$. Suppose $I$ is an $\mathfrak m$-primary ideal of $R$, i.e., $(x, y)^n \subset I \subset (x, y)$ for some $n$. If $I_{\...
- 55
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1
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287
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Ext modules of coherent sheaves and associated modules
Let $\mathcal{E}$ and $\mathcal{F}$ be two coherent sheaves on a polarized projective variety $(X,\mathcal{O}_X(1))$.
Denote by $E=\Gamma_*(\mathcal{E})=\oplus_{k\in\mathbb{Z}}\mathcal{E}(k)$,
$F=\...
- 2,444
1
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1
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330
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semilocal total quotient ring whose J(R) is not zero
I am interested in rings in which every non-unit is a zero divisor. Can you give me an example of such a ring that also has FINITELY many maximal ideals (semilocal), and whose Jacobson radical is not ...
- 21
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183
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relation between Min(R) and Min(R^)
Let $\hat{R}$ is $m$-adic completion of a local ring $(R,m)$.What is the relation between $Min R$ and $Min \hat{R}$. we know that $\hat{R}$ is faithfully flat $R$-module.
$Min R$=set of all minimal ...
- 418
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1
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921
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how to prove that localisation preserves Hom's [closed]
Can anyone tell me where I can read a proof that the natural map
$Hom_{A}(M,N)[S^{-1}]\rightarrow Hom_{A[S^{-1}]}(M[S^{-1}],N[S^{-1}])$
is an isomorphism if $M$ is finitely presented?
- 2,125
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1
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482
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Injective hulls of residue fields of a local ring and its ring invariants by finite group action
Let $R$ be a local ring, $m$ its maximal ideal and $k:= R/m$ its residue field.
Suppose that a finite cyclic group $G= \mathbb{Z}/ m \mathbb{Z}$ has a linear nontrivial action on $R$.
Let $R^G$ be a ...
- 909
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1
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403
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Prime ideals in univariate polynomial rings
I'm looking for a textbook reference of the following elementary fact (a reference for an excercise in a textbook is also welcome):
Let $R$ be a commutative ring and let $\mathfrak{p}$ be a prime ...
- 16.2k