Characterizing Maximal Ideals in Continuous Function Space

Elementary Algebraic Structures

Henri Bourlès , in Fundamentals of Advanced Mathematics, 2017

2.3.3 Maximal ideals and prime ideals. Spectrum

(I) Maximal ideals A submodule that is maximal (with respect to inclusion) among the proper submodules of an R-module M (section 2.3.1 (II)) is called a maximal submodule of M.

Definition 2.26

Let R be a ring. A maximal left ideal in R is a maximal submodule of _RR. If R is commutative, the set of maximal ideals in R is called the maximal spectrum of R and is written as Spm (R).

Theorem 2.27 Krull

1) Let M  ≠   0 be a finitely generated left R -module and let N be a proper submodule of M. Then, there exists a maximal submodule of M that contains N. 2) In particular, every proper left ideal in R  ≠   0 belongs to some maximal left ideal.

Proof.– (1) Let X be a generating set of M, $\mathcal{P}$ the set of proper submodules of M that contain N, and C a chain of $\mathcal{P}$ ordered by inclusion. The union A of the elements of C is a submodule of M. If A  = M, then A  ⊃ X and since X is finite, all the elements of X must belong to some element of $\mathcal{P}$ , which is impossible. Hence, A is a proper submodule of M, which implies that $\mathcal{P}$ is inductive, and so must have a maximal element by Zorn's lemma (Lemma 1.3). (2) is a special case of (1).

The following result, whose proof is an exercise* ([COH 03a], Thm. 10.2.4), establishes a useful generalization of Krull's theorem:

Lemma 2.28

Let R be a commutative ring, S  ⊂ R a multiplicative monoid and $\mathfrak{a}\ne \mathbf{R}$ an ideal disjoint from S. Then, there exists an ideal $\mathfrak{m}$ that is maximal among the ideals that contain $\mathfrak{a}$ and are disjoint from S and $\mathfrak{m}$ is a prime ideal.

(II) Prime spectrum of a ring.

Definition 2.29

A proper ideal $\mathfrak{p}⊲\mathbf{R}$ is said to be prime if, for all ideals $\mathfrak{a},\mathfrak{b}⊲\mathbf{R}$ ,

$\mathfrak{ab}\subset \mathfrak{p}\Rightarrow \mathfrak{a}\subset \mathfrak{p}\mathit{or}\mathfrak{b}\subset \mathfrak{p}.$

The set of all prime ideals in R is called the prime spectrum of R and is written as Spec (R).

The ideals in $\mathbb{Z}$ are principal, so are of the form (n). These ideals are proper if and only if n  ≠   1, and maximal if and only if n  >   0 is a prime number. The ideal (0) is prime but is not maximal.

By Krull's theorem (Theorem 2.27), Spec (R)   ≠   ∅ if R  ≠   {0}. The proof of the following result is left as an exercise* ([LAM 01], Prop. (10.2)):

Lemma 2.30

For a proper ideal $\mathfrak{p}⊲\mathbf{R}$ , the following conditions are equivalent:

i)

$\mathfrak{p}$ is prime;

ii)

For a, b ∈ R, $\left(a\right)\left(b\right)\subset \mathfrak{p}$ implies that $a\in \mathfrak{p}$ or $b\in \mathfrak{p}$ ;

iii)

For a, b ∈ R, $a\mathbf{R}b\subset \mathfrak{p}$ implies that $a\in \mathfrak{p}$ or $b\in \mathfrak{p}$ ;

iv)

For $\mathfrak{a}$ , $\mathfrak{b}{⊲}_l\mathbf{R}$ , $\mathfrak{ab}\subset \mathfrak{p}$ implies that $\mathfrak{a}\subset \mathfrak{p}$ or $\mathfrak{b}\subset \mathfrak{p}$ ;

v)

For $\mathfrak{a}$ , $\mathfrak{b}{⊲}_r\mathbf{R}$ , $\mathfrak{ab}\subset \mathfrak{p}$ implies that $\mathfrak{a}\subset \mathfrak{p}$ or $\mathfrak{b}\subset \mathfrak{p}$ .

An ideal $\mathfrak{p}⊲\mathbf{R}$ is said to be completely prime (or strongly prime) if the ring $\mathbf{R}/\mathfrak{p}$ is entire. This condition is satisfied if and only if $\mathfrak{p}\ne \mathbf{R}$ and, for all

, or alternatively if and only if $\mathfrak{p}\ne \mathbf{R}$ and $\mathit{ab}\in \mathfrak{p}\Rightarrow a\in \mathfrak{p}$ or $b\in \mathfrak{p}$ . A completely prime ideal is prime, and the converse holds if R is commutative.

Lemma 2.31

Let

be a regular and invariant element. Then, p is prime in R ( section 2.1.1 (II)) if and only if the principal ideal (p) is completely prime.

Proof.– Let c ∈ R. We have that p | c if and only if c ∈ (p), so c ∉ (p) if and only if p ∤ c. But p is prime if and only if (p ∤ a and p ∤ b)   ⇒ p ∤ ab; in other words

.

By applying Zorn's lemma (Lemma 1.3), it can be shown that every completely prime ideal $\mathfrak{p}$ in a ring R contains a minimal completely prime ideal in R (exercise).

(III) Zariski topology. A topology $\mathfrak{T}$ on a set X is a subset of $\mathfrak{P}\left(X\right)$ such that $\varnothing \in \mathfrak{T}$ ; $X\in \mathfrak{T}$ ; if (O _{i∈ I} ) is a family of elements of $\mathfrak{T}$ , then ${\cup }_{i\in I}{O}_i\in \mathfrak{T}$ ; if O ₁, …, O_n are (a finite number of) elements of $\mathfrak{T}$ , then their intersection is in $\mathfrak{T}$ . The elements of $\mathfrak{T}$ are called the open sets of the topological space $\left(X,\mathfrak{T}\right)$ (or of X whenever $\mathfrak{T}$ is implicit); the complement of an open set is called a closed set. A neighborhood of a point x is a set $\mathcal{V}$ that contains an open set containing x.

Let R be a commutative ring, let $\mathrm{\Im }\left(\mathbf{R}\right)$ be the set of ideals of R and, given an ideal $\mathfrak{a}$ , let $\mathrm{V}\left(\mathfrak{a}\right)$ be the set of prime ideals containing $\mathfrak{a}$ . If $\mathrm{\Im }\left(\mathbf{R}\right)$ and Spec(R) are ordered by inclusion, the mapping $\mathrm{\Im }\left(\mathbf{R}\right)\to \mathrm{Spec}\left(\mathbf{R}\right):\mathfrak{a}\to \mathrm{V}\left(\mathfrak{a}\right)$ is decreasing. For any index set I, we also have that:

$\begin{array}{llll}\mathrm{V}\left({\displaystyle {\sum }_{i\in I}{\mathfrak{a}}_i}\right)\hfill & =\hfill & {\displaystyle \bigcap _{i\in I}\mathrm{V}\left({\mathfrak{a}}_i\right),}\hfill & \mathrm{V}\left({\mathfrak{a}}_1\cap {\mathfrak{a}}_2\right)=\mathrm{V}\left({\mathfrak{a}}_1{\mathfrak{a}}_2\right)=\mathrm{V}\left({\mathfrak{a}}_1\right)\cup \mathrm{V}\left({\mathfrak{a}}_2\right),\hfill \\ \mathrm{V}\left(\left(0\right)\right)\hfill & =\hfill & \mathrm{Spec}\left(\mathbf{R}\right),\hfill & \mathrm{V}\left(\mathbf{R}\right)=\varnothing ,\hfill \end{array}$

so the $\mathrm{V}\left(\mathfrak{a}\right)$ satisfy the axioms of the closed sets of a topology on Spec (R), called the Zariski topology on Spec (R). This is a Kolmogorov space, which means that, given two distinct points in Spec (R), there exists a neighborhood of one that does not contain the other ([BKI 98], Chap. II, section 4, Exerc. 9); however, it is not Hausdorff (the Hausdorff separation axiom, stating that any two distinct points have disjoint neighborhoods, is stronger than the Kolmogorov separation axiom) ⁶ . A point $\mathfrak{a}$ is closed in Spec (R) if and only if $\mathfrak{a}$ is a maximal ideal in R (exercise).

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9781785481734500021

History of Homological Algebra

Charles A. Weibel , in History of Topology, 1999

4.2.2. Tor_*(k, k) for local rings.

Consider a local ring R with maximal ideal m and residue field k = R/m. Cartan and Eilenberg had shown that Tor ^R _*(k, k) was a graded-commutative k-algebra [41, XI.4 and XI.5]. Its Hilbert function is just the sequence of Betti numbers b _i = dim Tor ^R _i (k, k), and it is natural to consider the Poincaré–Betti series

$P_R(t)=\sum _{i=0}^{\infty }{b}_i{t}^i.$

Note that the first Betti number is b ₁ = dim(m/m²). For example, if R is a regular ring, it was well known that Tor ^R _* (k, k) was an exterior algebra, so that P _R (t) = (1 + t) ^{b ₁} .

Serre showed in 1955 [166] that one always had P _R (t) ⩾ (1 + t) ^{b ₁} , i.e. that b _i is at least $\left(\begin{array}{c}{b}_1\\ i\end{array}\right)$ . In particular, if i = b ₁ then b _i ⩾ 1 and so Tor ^R _{b 1} (k, k) ≠ 0. As we mentioned above, this was the key step in Serre's proof that local rings of finite global dimension are regular. In his 1956 study [184], Tate showed that k had a free R-module resolution F _* which was a graded-commutative differential graded algebra, and used this to show that if R is not regular then P _R (t) ⩾ (1 + t) ^{b ₁} /(1 – t ²), i.e. that b _i is at least $\left(\begin{array}{c}{b}_1\\ i\end{array}\right)+\left(\begin{array}{c}{b}_1\\ i-2\end{array}\right)+\cdots .$ . This is the best lower bound. In case R is the quotient of a regular local ring by a regular sequence of length r (contained in the square of the maximal ideal), Tate showed that the Poincaré–Betti series of R is the rational function P _R (t) = (1 + t) ^{b ₁} /(1 – t ²) ^r .

Based upon Tate's results, Serre stated in his lecture notes [169, p. 118] that it was not known whether or not P _R (t) was always a rational function. This problem remained open for over twenty years, until it was settled negatively in 1982 by David Anick [8]. Anick's example was an Artinian algebra R with m³ = 0. Constructing a finite simply-connected CW complex X whose cohomology ring was R, a 1979 result of Roos [158] showed that the Poincaré–Betti series of the loop space ΩX,

$H(t)=\sum \text{dim}{H}_i(\Omega X)t^i,$

was not a rational function either. This settled a second problem of Serre, also posed in [169, p. 118].

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444823755500298

An Introduction to Homological Algebra

In Pure and Applied Mathematics, 1979

Theorem 3.4

Every commutative ring R has IBN.

Proof

By Zorn's lemma, R has a maximal ideal M, whence R/M is a field. Let A be free with basis $\left\{a_i:i\in I\right\}$ . By Exercise 2.4, the quotient module A/MA is an R/M-module (being annihilated by M), i.e., A/MA is a vector space over R/M. Since $MA={\displaystyle \coprod M{a}_i}.$ we see that $A/MA\cong {\displaystyle \coprod \left(R{a}_i/M{a}_i\right).}$ . Since $R{a}_i/M{a}_i\cong R/M,$ it follows that A/MA has dimension card(I). If $\left\{b_j:j\in J\right\}$ is a second basis of A, the same argument gives dim A/MA = card(J). Since fields have IBN, we conclude that card(I) = card(J).

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S0079816908604493

Cyclic Codes

MinJia Shi , ... Patrick Sole , in Codes and Rings, 2017

7.3.1.3 Hamming Distance of the Codes

For $\mathbf{c}\in R^n$ we denote by $\mathrm{wt}(c)$ the Hamming weight of c, that is, the cardinality of $\mathrm{supp}(\mathbf{c})=\{i|c_i\ne 0\}$ , the support of c. The minimum Hamming weight of the nonzero elements in a code $\mathcal{K}\subseteq R^n$ , will be denoted by $d(\mathcal{K})$ . We show that under certain weak assumptions this is the actual minimum distance of the code, as in the case of classical linear codes.

Definition 7.31

Let R be a local ring with maximal ideal $M=\text{<mi mathvariant="italic" is="true">rad</mi>}(R)$ and residue field ${\mathbb{F}}_q=\bar{R}$ . The socle $S(\mathcal{K})$ of an R-linear code $\mathcal{K}$ is defined to be the sum of all its irreducible R-submodules.

According to [11], the equality

$S(\mathcal{K})=\{\mathbf{c}\in \mathcal{K}|M\mathbf{c}=0\}$

holds for any R-linear code $\mathcal{K}$ . So we may consider $S(\mathcal{K})$ as a linear space over the field ${\mathbb{F}}_q$ where $\overline{r}\cdot \mathbf{c}=r\mathbf{c}$ for all $\overline{r}\in {\mathbb{F}}_q$ , $\mathbf{c}\in S(\mathcal{K})$ .

Lemma 7.32

Let R be a local ring with maximal ideal M and $\mathcal{K}$ an R-linear code of length n. Then $S(\mathcal{K})$ is a linear code of length n over the field ${\mathbb{F}}_q=R/M$ and $d(\mathcal{K})=d(S(\mathcal{K}))$ .

Proof

It is a direct translation of Proposition 5 in [11]. □

Proposition 7.33

Assume the conditions of Theorem 7.29 . Then $d(\mathcal{K})=d(\mathcal{L})$ , where $\mathcal{L}$ is the code $〈\overline{G_1},\dots ,\overline{G_t}〉+\bar{I}$ in ${\mathbb{F}}_q[X_1,\dots ,X_r]/〈{\bar{t}}_1(X_1),\dots ,{\bar{t}}_r(X_r)〉$ .

Proof

The socle of the code $\mathcal{K}$ is $S(\mathcal{K})=〈a^{t-1}{G}_1,a^{t-1}{G}_2,\dots ,a^{t-1}{G}_t〉+I$ , which can be viewed as a linear code over ${\mathbb{F}}_q$ . Consider the ${\mathbb{F}}_q$ -vector space isomorphism $\varphi :a^{t-1}R[X_1,\dots ,X_r]/I\to {\mathbb{F}}_q[X_1,\dots ,X_r]/\bar{I}$ , given by $a^{t-1}g+I\to \bar{g}+\bar{I}$ to conclude the result. □

In the general situation we cannot state that the minimum distance of a semisimple code $\mathcal{K}$ is equal to the minimum distance of the code $\bar{\mathcal{K}}$ . The most we can say is that, if $\bar{\mathcal{K}}\ne 0$ , then $d(\mathcal{K})\le d(\bar{\mathcal{K}})$ . However, there is one subclass of multivariable semisimple codes for which the equality holds.

Definition 7.34

With the conditions of Theorem 7.29 , the code $\mathcal{K}$ is called a Hensel lift of a multivariable semisimple code if $〈G_1+I〉\ne I$ and $〈G_i+I〉=0$ , for all $i=2,\dots ,t$ .

This notion generalizes the definition of a Hensel lift of a cyclic code introduced in [12]. For this class of codes, we have the following result.

Corollary 7.35

If $\mathcal{K}\ne 0$ is a Hensel lift of a multivariable semisimple code, then $d(\mathcal{K})=d(\bar{\mathcal{K}})$ .

Proof

As noticed above the inequality $d(\mathcal{K})\le d(\bar{\mathcal{K}})$ holds. On the other hand, since $\mathcal{K}$ is a Hensel lift of a multivariable semisimple code, we have that $\mathcal{L}=\bar{\mathcal{K}}$ and the result follows from the previous proposition. □

This corollary generalizes [12, Corollary 4.3] for a Hensel lift of cyclic codes. Moreover, all classical bounds on distances for semisimple codes over fields (BCH, Hartmann–Tzeng, Roos, …) also apply to their Hensel lifts. We remark that these bounds can be stated in the multivariable Abelian case due to [15, Prop. 8, Chap. 6], covered in Proposition 7.37 below.

Definition 7.36

A multivariable semisimple code $\mathcal{K}\subset R[X_1,\dots ,X_r]/I$ is called Abelian, if $I=〈x_1^{e_1}-1,\dots ,X_r^{e_r}-1〉$ , where $e_1,\dots ,e_r\in \mathbb{N}$ .

Let $S={\bigcup }_{i=1}^l{\bigcup }_{j=1}^{s_i}C({\mu }^{(i,j)})$ be the set of defining roots of a semisimple Abelian code in ${\mathbb{F}}_q[X_1,\dots ,X_r]/\bar{I}$ , where $C({\mu }^{(i,j)})\in \mathcal{C}$ such that $p_{{\mu }^{(i,j)},1}=p_{{\mu }^{(k,l)},1}$ if and only if $i=k$ . Consider for each class $C({\mu }^{(i,j)})$ the polynomial:

$\begin{array}{cc}\hfill & \frac{{\overline{t}}_1(X_1)}{p_{{\mu }^{(i,j)},1}(X_1)}(\prod _{k=2}^r\frac{{\overline{t}}_k(X_k)}{p_{{\mu }^{(i,j)},k}(X_k)}\prod _{k=2}^r{\pi }_{{\mu }^{(i,j)},k}(X_2,\dots ,X_r))\hfill \\ \hfill & =\frac{{\overline{t}}_1(X_1)}{p_{{\mu }^{(i,j)},1}(X_1)}(F_{ij}(X_2,\dots ,X_r)).\hfill \end{array}$

Here $p_{{\mu }^{(i,j)},k}$ for $k=1,\dots ,r$ , and ${\pi }_{{\mu }^{(i,j)},k}$ for $k=2,\dots ,r$ are as in Definition 7.15, and $F_{ij}\in {\mathbb{F}}_q[X_2,\dots ,X_r]$ is uniquely determined by the class $C({\mu }^{(i,j)})$ . Let us consider the field ${\mathbb{F}}^{(i)}={\mathbb{F}}_q(X_1)/p_{{\mu }^{(i,1)},1}(X_1)$ , and the code $J_i$ generated by ${\sum }_{j=1}^{s_i}{F}_{ij}$ in the algebra ${\mathbb{F}}^{(i)}[X_2,\dots ,X_r]/〈{\overline{t}}_2,\dots ,{\overline{t}}_r〉$ , $i=1,\dots ,l$ .

Proposition 7.37

With the above notation, the minimum weight of a semisimple code over a field ${\mathbb{F}}_q$ and of the corresponding Hensel lift over R is at least ${\min }_{1⩽i⩽l}\{d_i\cdot {\delta }_i\}$ , where $d_i$ is the minimum weight of the code in ${\mathbb{F}}_q[X_1]/\overline{t}(X_1)$ generated by

$\frac{\overline{t}(X_1)}{p_{{\mu }^{(i,1)},1}(X_1)\cdots p_{{\mu }^{(l,1)},1}(X_1)}$

and ${\delta }_i$ is the minimum weight of the code $J_i$ .

Proof

It is a straightforward generalization of Lemma 7.32 and [15, Prop. 8, Chap. 6]. □

Remark 7.38

Notice that, in view of this result, the computation of the minimum distance of a semisimple Abelian code in r variables is reduced to computations of minimum distances of semisimple Abelian codes in fewer number of variables.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128133880000070

Quasicyclic Codes

MinJia Shi , ... Patrick Sole , in Codes and Rings, 2017

8.2.1.1 Rings

A ring A is local if it admits a unique maximal ideal M. In that case the quotient ring $k:=A/M$ is a field. Factorizations fg of elements h of $k[X]$ can be "lifted" to factorizations FG of H in A in such a way that $f,g,h$ correspond to $F,G,H$ , respectively, under reduction modulo M. This is the so-called Hensel lifting. For the special case of $A={\mathbf{Z}}_4$ , so $k={\mathbb{F}}_2$ ; see, for instance, [3].

A ring is a chain ring if and only if it is both local and principal. A local ring is a chain ring if and only if its maximal ideal has a unique generator t, say, $M=(t)$ . With this notation, the ideals of A constitute a chain for inclusion

$A\supset (t)\supset (t^2)\supset \cdots \supset (t^{d-1})\supset (t^d)=(0).$

The integer d is then called the depth of A. If k, as a finite field, has q elements, then $A/(t^i)$ has $q^i$ elements, so A has $q^d$ elements.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128133880000082

Functional Analysis and its Applications

Wieslaw Żelazko , in North-Holland Mathematics Studies, 2004

Theorem 9

Let A be a real or complex unital F-algebra. Then A has all maximal two-sided ideals closed iff it is a Q₂-algebra.

As before, we do not know whether there exists a Q ₂-algebra which is not a Q-algebra, but we suspect that such an algebra can exist.

For an infinite dimensional commutative F -algebra it can happen that not only maximal ideals are closed, but also all ideals are closed (in the Banach algebra case such a situation cannot exist, as follows from the mentioned below result of Grauert and Remmert [15]). For instance, consider the algebra A of all formal power series $x={\displaystyle {\sum }_0^{\infty }{\xi }_{\iota }{t}^{\iota }}$ with usual multiplication (convolution multiplication of coefficients) and the seminorms

${\left|x\right|}_n={\displaystyle \sum _0^n\left|{\xi }_i\right|}.$

As a t.v.s it coincides with the space (s) considered by Banach in [8] (example 2 in §7 of the Introduction). Clearly these seminorms satisfy (2), so that s is a commutative m-convex B_o -algebra. It is not hard to see that every proper ideal of A is of the form I_n   = tⁿA, n  =   1, 2,… (it follows from the fact that every element x in A, for which ξ ₀ ≠ 0, is invertible in A, in particular A is a Q-algebra), and so it is closed. The characterization of commutative F-algebras with all ideals closed was obtained independently by Choukri and El Kinani [12], and the author [37].

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S0304020804801804

Higher Algebraic K-Theory

Aderemi Kuku , in Handbook of Algebra, 2006

4.3 Devissage

4.3.1. Devissage theorem [114]. Let $\mathcal{A}$ be an Abelian category, $\mathcal{B}$ a non-empty full subcategory closed under subobjects, quotient objects and finite products in $\mathcal{A}$ . Suppose that every object M of $\mathcal{A}$ has a finite filtration 0 = M ₀ ⊂ M ₁ ⊂ ⋯ ⊂ M_n = M such that $M_i/M_{i-1}\in \mathcal{B}$ for each i, then the inclusion $Q\mathcal{B}\to Q\mathcal{A}$ is a homotopy equivalence. Hence $K_i\left(\mathcal{B}\right)\cong K_i\left(\mathcal{A}\right)$ .

4.3.2. Corollary [114]. Let $\underset{¯}{a}$ be a nilpotent two-sided ideal of a Noetherian ring R. Then for all n ≥ 0, $G_n\left(R/\underset{¯}{a}\right)\cong G_n\left(R\right)$ .

4.3.3. Examples.

(i)

Let R be an Artinian ring with maximal ideal $\underset{¯}{m}$ such that ${\underset{¯}{m}}^r=0$ for some r. Let $k=R/\underset{¯}{m}$ (e.g., $R\equiv \mathbb{Z}/p^r$ , $k\equiv {\mathbb{F}}_p$ ). In 4.3.1 put $\mathcal{B}$ = category of finite-dimensional k-vector spaces and $\mathcal{A}=\mathcal{M}\left(R\right)$ . Then we have a filtration $0={\underset{¯}{m}}^{r}M\subset {\underset{¯}{m}}^{r-1}M\subset \cdots \subset \underset{¯}{m}M\subset M$ for any $M\in \mathcal{M}\left(R\right)$ . Hence by 4.3.1, G_n (R) ≈ K_n (k).

(ii)

Let X be a Noetherian scheme, i: Z ⊂ X the inclusion of a closed subscheme. Then $\mathcal{M}\left(Z\right)$ is an Abelian subcategory of $\mathcal{M}\left(X\right)$ via the direct image $i:\mathcal{M}\left(Z\right)\subset \mathcal{M}\left(X\right)$ . Let ${\mathcal{M}}_Z\left(X\right)$ be the Abelian category of O_X -modules supported on Z, $\underset{¯}{a}$ an ideal sheaf in O_X such that $O_X/\underset{¯}{a}=O_Z$ . Then every M ∈ M_Z (X) has a finite filtration $M\supset M\underset{¯}{a}\supset M{\underset{¯}{a}}^2\supset \cdots$ and so, by devissage, $K_n\left({\mathcal{M}}_Z\left(X\right)\right)\approx K_n\left(\mathcal{M}\left(Z\right)\right)\approx G_n\left(Z\right)$ .

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S1570795406800045

An Introduction to Homological Algebra

In Pure and Applied Mathematics, 1979

Corollary 11.27

Let R be a ring with right ideal I and left ideal J. Then

(i)

$\begin{array}{ccc}{\text{Tor}}_n\left(R/J,R/J\right)\cong {\text{Tor}}_{n-2}\left(I,J\right)& for& n>2;\end{array}$

(ii)

${\text{Tor}}_2\left(R/J,R/J\right)\cong \ker \left(I\otimes J\to IJ\right);$

(iii)

${\text{Tor}}_1\left(R/I,R/J\right)\cong \left(I\cap J\right)/IJ.$

Proof

This follows immediately from the Theorem applied to the exact sequences $0\to I\to R\to R/I\to 0$ and $0\to J\to R\to R/J\to 0$ .

As an instance of this, let R be a commutative local ring with maximal ideal m and residue field $k=R/m$ . Then ${\text{Tor}}_1^R\left(k,k\right)\cong m/m^2$ . Of course, one may prove this more simply.

Proof of Theorem 11.26

(i)

Theorem 11.23 gives an anticommutative diagram of connecting homomorphisms:

If n > 2, these maps are isomorphisms, for all the bordering terms are "honest" Tor's (i.e., not Tor₀) which vanish because a variable is flat. Therefore, for n > 2

${\text{Tor}}_n\left(A,B\right)\cong {\text{Tor}}_{n-2}\left(K,L\right).$

(ii)

There is a diagram with exact rows and columns that is almost commutative (the upper left square is only anticommutative):

It follows that

$\begin{array}{llll}{\text{Tor}}_2\left(A,B\right)\cong {\text{Tor}}_1\left(A,L\right)\hfill & \cong \hfill & \ker \left(\alpha \otimes 1_L\right)\hfill & \hfill \\ \hfill & =\hfill & \ker \left(\gamma \left(\alpha \otimes 1_L\right)\right),\hfill & \text{since}\gamma \text{ is monic,}\hfill \\ \hfill & =\hfill & \ker \left(\alpha \otimes \beta \right).\hfill & \hfill \end{array}$

(iii)

Apply Lemma 11.25 to the commutative diagram

Since P and Q are flat, the maps $i=1_p\otimes \beta$ and $j=\alpha \otimes 1_Q$ are monic. But the map $\overline{j}:\text{coker}\left(1_K\otimes \beta \right)\to \text{coker}\left(1_p\otimes \beta \right)$ is just $\alpha \otimes 1_B$ , whose kernel is ${\text{Tor}}_1\left(A,B\right)$ . The final part of the Lemma gives the desired isomorphism.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S0079816908604572

Probability on MV-Algebras

Beloslav Rieĕan , Daniele Mundici , in Handbook of Measure Theory, 2002

THEOREM 1.18

Let M be a σ-complete divisible MV-algebra.

(i)

Let $\Omega =ℳ\left(M\right)$ be the compact Hausdorff space of maximal ideals of M. Then the map a ↦ a ^* of Theorem 1.5 is an isomorphism between M and the MV-algebra C(Ω) of all continuous [0, 1]-valued functions over Ω.

(ii)

Let F be the set of all functions f : Ω → [0, 1] which are equal to a continuous function g^* ∈ C(Ω) except for a meager set. Then g^* is uniquely determined by f, F is a full tribe over Ω, and the map f ↦ g^* ↦ g is a σ-homomorphism of F onto M.

The proof depends on the following

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444502636500221

Abstract C⁎-Algebras

In C*-Algebras and their Automorphism Groups (Second Edition), 2018

1.1.7 Theorem

If A is a commutative $C^{⁎}$ -algebra, then the Gelfand transform is a ⁎-preserving isometry of A onto $C_0(\hat{A})$ .

Proof

If $t\in \hat{A}$ and $x\in A$ , then kert is a maximal ideal of A, whence $t(x)\in \mathrm{Sp}(x)$ (and conversely, if $\lambda \in \mathrm{Sp}(x)\backslash \{0\}$ , then $\lambda =t(x)$ for some t in $\hat{A}$ ). Therefore, if $x=x^{⁎}$ , then $t(x)\in \mathsf{R}$ by 1.1.5. It follows that $t(x^{⁎})=\overline{t(x)}$ for each x in A, which shows that the map $x\to \hat{x}$ is ^⁎-preserving (using complex conjugation of functions as involution in $C_0(\hat{A})$ ). Moreover, $\Vert \hat{x}\Vert$ is the spectral radius of x, whence $\Vert \hat{x}\Vert =\Vert x\Vert$ by 1.1.4 as each x in A is normal. Thus $x\to \hat{x}$ is a ^⁎-preserving isometry of A into $C_0(\hat{A})$ , and since the set of functions $\{\hat{x}|x\in A\}$ separates points in $\hat{A}$ and does not vanish at any point, we conclude from the Stone–Weierstrass theorem that the image of A is $C_0(\hat{A})$ . □

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128141229000015

riddleoperepien.blogspot.com

Source: https://www.sciencedirect.com/topics/mathematics/maximal-ideal

Share this post