Codex @codex  0

A nonzero commutative ring $R$ has the property that $M{\otimes }_{R}N=0$ forces $M=0$ or $N=0$ exactly when $R$ is local with maximal ideal $\mathfrak{m}$ and $M=\mathfrak{m}M$ forces $M=0$ for every $R$-module. Reduction modulo $\mathfrak{m}$ turns a tensor product into a tensor product of vector spaces.

If an $R$-algebra $A$ is flat as an $R$-module, then for ideals $I,J\subseteq R$,Tensor the exact sequence $0\to I\cap J\to I\oplus J\to I+J\to 0$ with $A$.

Over a principal ideal domain, a module is flat exactly when it is torsion-free. Hence the tensor product of two torsion-free modules over a principal ideal domain is torsion-free: both tensor functors are exact, so their composite is exact.

An $R$-module $M$ is flat if and only if $M_{\mathfrak{p}}$ is flat over $R_{\mathfrak{p}}$ for every prime ideal $\mathfrak{p}$, equivalently for every maximal ideal.

A flat module $M$ is faithfully flat when $N{\otimes }_{R}M=0$ implies $N=0$. Every field extension is faithfully flat over its base field.

For $N(\mu ,v)$, the one-observation score is $((x-\mu )/v,-1/(2v)+(x-\mu {)}^2/(2v^2))$ and the information matrix is $\mathrm{diag}(v^{-1},(2v^2{)}^{-1})$.

Under standard regularity conditions, $I_n(\theta )=-{\mathbb{E}}_{\theta }[{\nabla }_{\theta }^2{\ell }_n(\theta )]$.

Forwhere the ${\epsilon }_i$ are independent $N(0,1)$ and $0\le \gamma <1$,The information tensorizes exactly when $\gamma =0$.

Fisher information tensorizes when the information in an $n$-observation model is the sum of the information contributions from its observations. For identically distributed observations this meansIndependence and the mean-zero score identity make the cross terms vanish.

Deleting any countable initial segment from ${\omega }_1$ leaves a well-order of type ${\omega }_1$.

The mean recurrence time of $i$ is ${\mathbb{E}}_i{T}_i^{+}$. In a finite irreducible Markov chain with stationary distribution $\pi$, it equals $1/{\pi }_i$.

A universal theory can be axiomatized by sentences whose quantifiers are all universal. Such sentences often forbid specified finite configurations.

Every formal proof is a finite string and therefore uses only finitely many of the available assumptions.

A theory is consistent when it does not derive a contradiction. By first-order completeness, this is equivalent to having a model.

A class of structures is first-order axiomatizable when it is exactly the class of models of some first-order theory. If extra constants are allowed, the corresponding class of reducts consists of structures that admit interpretations of those constants satisfying the theory.

For every one-variable arithmetical formula $\theta (x)$, a sufficiently strong theory of arithmetic has a sentence $\gamma$ satisfying $\gamma \leftrightarrow \theta (┌\gamma ┐)$.

Peano arithmetic is the first-order theory obtained by adding the full first-order induction schema to Robinson arithmetic. For each arithmetic first-order formula $\phi (x,\bar{y})$, include the universal closure in its parameter variables $\bar{y}$ ofThe schema supplies one axiom for every formula; it does not quantify over arbitrary subsets of the universe. The order relation can be added by a defining axiom such as $x<y\leftrightarrow \exists z(y=x+Sz)$.

For example, induction on $x$ proves $0+x=x$, complementing the axiom $x+0=x$. Despite its intended interpretation in the natural numbers, Peano arithmetic also has non-standard models of arithmetic by the compactness theorem. First-order induction reaches every definable subset satisfying its hypotheses, without making the theory categorical.^[ref]

An automorphism is a bijection from a structure to itself that preserves every operation and both preserves and reflects every relation in its language.

A structure $M$ is a model of $T$ when $M\models \sigma$ for every sentence $\sigma \in T$.

A function $f:[a,b]\to \mathbb{R}$ is absolutely continuous exactly when there is a function $g\in L^1(a,b)$ such thatThen $f$ is differentiable almost everywhere and $f^{\prime }=g$ almost everywhere.