1
vote
1answer
51 views

A Direct Sum of Members of a Certain Class of Modules

Let $S$ be a class of $R$-modules and let an $R$-module $M$ be countably generated. Suppose that, for every direct summand $K$ of $M$, each element of $K$ belongs to a direct summand of $K$ that is ...
1
vote
0answers
33 views

A condition of equivalence of flatness and projectiveness

This is a problem in "Foundations of Module and Ring Theory" of Wisbauer: " Let $R$ be a subring of the ring $S$ containing the unit of $S$. Show that a flat $R$-module $N$ is projective if and only ...
2
votes
0answers
66 views

Direct product of direct sum of a flat module

I have a problem concerning flat modules: Let $M$ be an $R$-module such that the direct product $M^A$ is flat for all sets $A$. I want to prove that $(M^{(B)})^A$ is also flat for any sets $A$ ...
0
votes
1answer
43 views

Localization of a direct product

Is the localization of a direct product of two rings at a maximal (or prime) ideal identified with a localization of one of them? I would appreciate for any detailed answer.
1
vote
0answers
28 views

An inverse limit of a certain inverse system

Let $∆$ be a directed set and $(N_i,f_{ji})_{i∈∆}$ be an inverse system of $R$-modules. Fix $α \in∆$ and consider $(M_i,g_{ji})_{i\in∆}$ as follows: $M_i=N_i$ for $i≥α$, $M_i=0$ for $i<α$, and ...
1
vote
0answers
37 views

Direct limits commute with $\mathrm{Tor}$ functor

How one could prove that direct limits commute with the functor $\mathrm{Tor}$? Of course, I know that $\mathrm{Tor}$ with its first $0$ index is the same as tensor product which does commute ...
1
vote
0answers
27 views

Any counterexample for inverse limit functor not to be right exact [duplicate]

We know that inverse limit is a "left" exact functor on the category of modules in the sense that whenever $r:(A_i,α_j^i )→(B_i,β_j^i )$ and $s:(B_i,β_j^i )→(C_i,γ_j^i )$ are transformations of ...
0
votes
2answers
55 views

Injective resolution for an integral domain

How could one write an injective resolution for an arbitrary commutative integral domain $R$? Thanks in advance!
1
vote
1answer
55 views

A statement equivalent to flatness

If $R$ is a ring with identity and $P$ is a flat right $R$-module, it is a fact that any $R$-homomorphism $f$ from a finitely presented right $R$-module $M$ to $P$ factors through a finitely generated ...
0
votes
1answer
32 views

Examples of algebras having a module basis

I'm looking for examples of associative $R$-algebras, for which an $R$-module basis can be specified. Of course, if $K$ is a field, then any $K$-algebra admits such a basis, but this dis not what I'm ...
0
votes
0answers
52 views

Characteristic of ring $R_1\otimes\ldots\otimes R_k$

Let $R_1,\ldots, R_k$ be unital rings and $\otimes=\otimes_\mathbb{Z}$ and $\mathrm{chr}$ the characteristic. How can one see that $$\mathrm{chr}(R_1\otimes\ldots\otimes ...
2
votes
2answers
78 views

divisible modules

In surveying LMR of T.Y.Lam, I reached a paragraph stating that "when R is a domain every direct sum or direct product of divisible modules is divisible." My question is that "should R is not a ...
1
vote
0answers
28 views

Grading and commutators

If $R$ is a unital associtaitve commutative ring, then can any $R$-algebra $A$ may be filtered as $A_0:=A$ and $A_{i+1}:=(A_i,A_i)$ where $(-,-)$ is the commutator of $A$ with respect to $A$'s ...
3
votes
1answer
59 views

tensor, symmetric, exterior power of a module over a PID

Let $R$ be a PID and $M\cong R^r\!\oplus\bigoplus_{i=1}^s\!R/Ra_i$. Denote the tensor, symmetric, exterior power of $M$ by $T^nM=\bigotimes_{k=1}^nM$ and $S^nM= T^nM/\langle ...
5
votes
2answers
92 views

if $A^\times $ is a commutative group, does $A$ necessarily be a commutative ring?

Let $A$ be a ring and $A^\times$ be the collection of unit elements of $A$. If $A$ is a commutative ring, then $A^\times$ is a commutative group. Conversely, if $A^\times $ is a commutative group, ...
1
vote
0answers
59 views

Is the universal enveloping algebra of the free Poisson algebra generated by finite set (left) noetherian?

Let $P$ be the free Poisson algebra over $k$ (a field) generated by finite set $x_1,...,x_n$. Let's consider the universal enveloping algebra $P^e$ of the free Poisson algebra $P$. Hence a Poisson ...
0
votes
1answer
28 views

Show that a subalgebra is commutative.

If $B$ is an unital algebra (even not commutative), how do I show that the subalgebra spanned by the elements $1$, $f$ and $(f - \lambda1)^{-1}$ is commutative? Thank you.
0
votes
0answers
28 views

Integral closure [duplicate]

How can we see if $S=Z[x]$ and let R be a subring generated by 2x.then x is not integral over R. by definition if x is not integral over R then it means. There does not exist any element n such ...
6
votes
1answer
76 views

A finite unital and commutative ring with exactly one maximal ideal has $p^{n}$ elements.

Suppose $R$ is a finite unital and commutative ring that has exactly one maximal ideal. Prove that $\left | R \right |=p^{n}$ where $p$ is a prime number. If $R$ will be non-commutative, do we have ...
2
votes
2answers
86 views

Is $R$ initial in the category of $R$-algebras?

Let $R$ be an arbitrary unital associative ring. In the category of $R$-algebras $\mathfrak{Alg_R}$, if we consider $R$ as an $R$-algebra over itself (trivially), what type of object is it then in ...
4
votes
1answer
57 views

Weak flat condition?

Let $R$ be a unit ring (not necessarily commutative). Then it is clear that for a right $R$-module $M$ we have: $M$ is flat $R$-module $\Rightarrow$ for any left $R$-module $E$ with $E\otimes_{R}M=0$ ...
8
votes
3answers
816 views

Commutative property of ring addition

I have a simple question answer to which would help me more deeply understand the concept of (non)commutative structures. Let's take for example (our teacher's definition of) a ring: Let $R\neq ...
4
votes
2answers
419 views

Definition of a filtration on a ring, module, algebra

In most books, a graded ring/module/algebra means either a $\mathbb{N}$- or $\mathbb{Z}$-graded ring/module/algebra. But often, different gradings appear: doubly graded (spectral sequences) = ...
7
votes
1answer
144 views

Projective objects in the category of rings

What are the projective objects in the category of rings with identity ? Remarks: The only projectives I could find so far are $\{ 0\}$ and $\mathbb{Z}$. If $R$ is projective and ...
3
votes
1answer
108 views

tensoring a polynomial algebra: $R[x_i]/\mathfrak{a}\otimes A \cong A[x_i]/\mathfrak{a}$

Let $R$ be a commutative unital ring, $A$ an associative unital $R$-algebra, $I$ an arbitrary set, and $\mathfrak{a}$ an ideal of $R[x_i; i\!\in\!I]$. If $A$ is commutative, then there is an ...
2
votes
0answers
81 views

Non-commutative integral extensions?

In Commutative algebra there is a notion of an integral extension: Let $P$ be a subring of $R$. Then $R$ is the integral extension of $P$ if each element of $R$ is a root of a monic polynomial with ...
3
votes
0answers
93 views

Applications of Govorov-Lazard Theorem?

The Govorov-Lazard Theorem states that a (right) module over an unital ring is flat iff it is a direct limit of finitely generated free (right) modules. I wonder if this theorem has interesting ...
3
votes
1answer
193 views

An $(R,S)$-bimodule is a left $R \otimes_k S^{\text{op}}$-module

Let $k$ be a commutative ring, and let $R,S$ be $k$-algebras. To me "$R$ is a $k$-algebra" means that $R$ is a $k$-module such that $a(rs)=(ar)s=r(as)$ for all $a\in k$ and $r,s \in R$. Let $M$ be a ...
1
vote
1answer
288 views

Annihilator of a simple module

Let $R$ be a finitely generated commutative ring and $C$ an $R$-algebra ($C$ is not necessarily commutative). Assume that $C$ is a finitely generated $R$-module. If $S$ is a simple $C$-module, then ...
3
votes
0answers
86 views

Computation of determinant of a matrix with elements from an arbitrary commutative ring

The cofactor formula for computing the determinant of a matrix is applicable when elements of the matrix are from a commutative ring. However, the complexity of this method is extremely high and I ...
1
vote
1answer
236 views

Finding all simple $R$ modules of a ring.

I was hoping someone had an idea on how to go about solving the following; Find (up to isomorphism) all simple R-modules where i) $R = \begin{pmatrix} \mathbb{Z}/15 \mathbb{Z} & \mathbb{Z}/15 ...