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Suppose the additive group of the integral domain $R$ is isomorphic to ${\mathbb{Z}}^n$. Every ideal $I$ is then an additive subgroup of a finitely generated free abelian group, so it has finitely many additive generators $x_1,\dots ,x_m$. These also generate $I$ as an $R$-ideal: integer coefficients are coefficients from the canonical copy of $\mathbb{Z}$ in $R$. Thus every ideal is finitely generated, proving that $R$ is Noetherian by noetherianity from finite additive rank.

For an example that fails (i)--(iv), takeIt is an integral domain and has additive group ${\mathbb{Z}}^2$. The idealhas index two: modulo $I$, one has $2=0$ and $\sqrt{-5}=-1=1$, and the resulting quotient is ${\mathbb{F}}_2$. If $I=(a+b\sqrt{-5})$, multiplication by its generator would have determinant and absolute indexwhich has no integer solution. Hence $I$ is not principal, as detailed in nonprincipal ideal in the integers adjoined a square root of minus five. Condition (i), and therefore all four equivalent conditions, fails.

The minimum-cost flow problem chooses, for each directed edge $(i,j)\in E$, a flow $x_{ij}$ and solvessubject toFeasibility requires ${\sum }_i{b}_i=0$.

Set $y_{ij}=x_{ij}-M_{ij}$. Thenand its balance vector isThe objective becomes $\sum C_{ij}{y}_{ij}$ plus the constant $\sum C_{ij}{M}_{ij}$. Translation by $M$ is a bijection between feasible flows and preserves their ordering by cost, so the transformed zero-lower-bound problem is equivalent.

For a regular constraint $g=0$, an interior constrained extremum satisfiestogether with the constraint; boundary cases must also be checked.

Here maximize $f=xyz$ subject to $x^2+y^2+z^2=L^2$. The Lagrange multiplier equations giveAt a positive maximizer these imply $x=y=z=L/\sqrt{3}$. Boundary points have zero volume, so the maximum is

A topological surface is a Hausdorff, second-countable space in which every point has a neighbourhood homeomorphic to an open subset of ${\mathbb{R}}^2$.

The antipodal action on $S^2$ is free. The surface quotient by a free finite action applies: Around each point choose a small open disc disjoint from its antipodal image; the quotient map restricts to a homeomorphism from that disc onto an open neighbourhood in the quotient. Compactness gives Hausdorffness and second countability descends from the sphere. Thus the quotient is the real projective plane, in particular a topological surface.

For the second quotient write a point away from the poles asThe mapin these cylindrical coordinates extends continuously over the poles and identifies exactly $\varphi$ with $\varphi +\pi$. It therefore induces a continuous bijection $S^2/\approx \to S^2$. The domain is compact and the sphere Hausdorff, so this bijection is a homeomorphism.

Choose bases of $V$ and $W$, of dimensions $v$ and $w$. A linear map is uniquely determined by the arbitrary images of the $v$ basis vectors, each with $w$ coordinates, so

Put $a=\dim A$ and $b=\dim B$, and extend bases of $A,B$ to bases of $V,W$. The condition $\varphi (A)\subseteq B$ forces the $(w-b)a$ matrix entries from $A$ to a complement of $B$ to vanish and imposes no other restriction. Hence

For the last part let $r=\dim (S\cap T)=s+t-v$. ChooseA map in $Y$ sends $S\cap T$ into itself, $S_0$ into $S$, and $T_0$ into $T$, with these choices independent. Therefore