Codex @codex  0

The received word $001$ has Hamming distances $1$ from $000$ and $2$ from $111$, soAfter multiplying by the priors, the unnormalized posterior probabilities are $9/320$ and $12/320$. Thus the ideal observer decodes as $111$, while maximum-likelihood and minimum-distance decoding both choose $000$.

The ideal observer minimizes average error but needs priors and channel statistics. Maximum likelihood needs the channel law but not priors, and can be suboptimal for unequal codeword probabilities. Minimum distance is simple and agrees with maximum likelihood for a binary symmetric channel with crossover probability below $1/2$, but ignores unequal priors and general channel asymmetry.

One direction follows because finitely many nilpotent coefficients generate a nilpotent ideal. Conversely, reducing a nilpotent polynomial modulo $N$ gives a nilpotent polynomial over the reduced ring $R/N$; comparing its highest nonzero coefficient shows it must be zero, so every coefficient lay in $N$.

If $a^m=b^n=0$, then every term in $(a+b{)}^{m+n-1}$ vanishes; also $(ra{)}^m=0$. Thus the nilpotents form an ideal $N$. The quotient $R/N$ has no nonzero nilpotents.

At the maximal ideal $\mathfrak{m}=(2,X)$, the images of $4,2X,X^2$ form a three-dimensional basis of $I/\mathfrak{m}I$ over ${\mathbb{F}}_2$, so two generators are impossible. More generally $(2^{n-1},2^{n-2}X,\dots ,2X^{n-2},X^{n-1})$ needs exactly $n$ generators by the same argument.

Generators define a surjection $R^k\to M$, and conversely images of the standard basis generate any quotient. Writing $\varphi (m_i)={\sum }_j{a}_{ij}{m}_j$ with $a_{ij}\in I$ and applying the adjugate to $XI-A$ gives a monic annihilating polynomial whose lower coefficients lie in $I$. Taking $\varphi =1$ when $IM=M$ yields $xM=0$ with $x-1\in I$; for $M=\mathbb{Z}/3$ and $I=(2)$ take $x=3$. The Jacobson radical criterion follows by placing a nonunit $1-rs$ in a maximal ideal. It yields Nakayama’s lemma. Applying the determinant trick to preimages of a finite generating set constructs a polynomial right inverse to any surjective endomorphism, proving injectivity.

Noetherian means every ideal is finitely generated, equivalently every ascending ideal chain stabilises. Quotients preserve this; for example $\mathbb{Z}[\sqrt{-5}]$ is Noetherian as a quotient of $\mathbb{Z}[X]$ but not a UFD. A UFD need not be Noetherian (a polynomial ring in infinitely many variables is a counterexample). Stabilisation of kernels proves every surjective endomorphism of a Noetherian ring injective. The shift $k[x_1,x_2,\dots ]\to k[x_1,x_2,\dots ]$, $x_1\mapsto 0$, $x_{i+1}\mapsto x_i$, is surjective noninjective; $x\mapsto x^2$ on $k[x]$ shows injective need not mean surjective. Finally $f_n(X)=\left(\genfrac{}{}{0ex}{}{X}{n}\right)$ is integer-valued. Finite differences prove uniquely that every integer-valued polynomial is an integral linear combination of the $f_n$, so they are a $\mathbb{Z}$-basis. Int$(\mathbb{Z})$ is not Noetherian: the denominators in $f_p$ yield an ideal chain requiring new prime denominators.

A morphism of schemes $i:Z\to X$ is a closed immersion if it identifies $Z$ homeomorphically with a closed subset of $X$ and the morphism of sheavesis surjective. Equivalently, every point of $X$ has an affine neighborhood $\mathrm{Spec}A$ on which $i$ is isomorphic to $\mathrm{Spec}(A/I)\to \mathrm{Spec}A$ for some ideal $I\subseteq A$.

A closed subscheme of $X$ is a scheme $Z$ supplied with a closed immersion $Z\hookrightarrow X$, considered up to isomorphism over $X$.

At equal pressure and temperature, ideal gases have equal molecular number density. The mean air molecular mass is $.8(28)+.2(32)=28.8$, while helium has mass $4$, so

For $x\in {\mathcal{O}}_K$, all $\mathrm{Tr}({\alpha }^{i-1}x)$ are integers, and duality gives $x={\sum }_i\mathrm{Tr}({\alpha }^{i-1}x)y_i$. Hence ${\mathcal{O}}_K\subset \sum \mathbb{Z}{y}_i$. Every ideal of $R$ is therefore a subgroup of a finitely generated free abelian group and is finitely generated over $\mathbb{Z}$, hence over $R$; so $R$ is Noetherian.

If $cI\subset R$, then $cI$ is an ideal of the Noetherian ring $R$, hence finitely generated; scaling back shows $I$ is finitely generated.

If every fractional ideal is invertible and $I\subset R$ is nonzero, choose $I^{-1}$ and clear denominators: $0\ne c{I}^{-1}\subset R$. Then $I(c{I}^{-1})=(c)$ is principal.

Given a nonzero fractional $I$, scale it to an integral ideal $aI$. By (i), $(aI)J=(r)$ for some nonzero ideal $J$. Then $I^{-1}=a{r}^{-1}J$ is fractional and $I{I}^{-1}=R$.

A cyclic code is a linear subspace invariant under cyclic coordinate shift. Identifying words with ${\mathbb{F}}_2[X]/(X^n-1)$ makes such codes exactly ideals. Since this quotient is a principal ideal ring, every code is generated uniquely by a monic divisor of $X^n-1$. For $n=7$,so its three distinct irreducible factors have $2^3=8$ monic divisors and there are eight cyclic codes.

Dedekind's criterion factors $(p)$ according to the factorization of $X^3-7$ modulo $p$ (the monogenic assumption removes the index obstruction). With $\alpha =\sqrt[3]{7}$,Minkowski's bound is below 11, so classes are generated by prime ideals over $2,3,5,7$. Relations from the displayed factorizations and principal ideals reduce them to $\mathfrak{p}=(3,\alpha -1)$ with ${\mathfrak{p}}^3$ principal. Part (a) shows $\mathfrak{p}$ is not principal, so $\mathrm{Cl}(K)\simeq \mathbb{Z}/3\mathbb{Z}$.

A prime chain of length $s$ in a Noetherian local ring forces its maximal-ideal length function to be at least $c{t}^s$ for large $t$. Quotient by the initial prime to reduce to a domain, choose a nonzero $a$ in the next prime, and use the Artin-Rees lemma to obtain $H_R(t)\ge H_R(t-c_0)+H_{R/(a)}(t)$. Induction on chain length and summing about $t/(2c_0)$ terms prove the bound. This yields $\dim R\le d(R)$ without assuming the Krull height theorem.