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An endemic equilibrium is an equilibrium at which infection persists, so at least one infected compartment is positive.

A disease-free equilibrium is an equilibrium of an epidemic model at which every infected compartment is zero.

A spatial SIR-type model combines compartment transitions with diffusion in the compartments whose members move appreciably.

Under mass action, random pairwise encounters between populations of sizes $X$ and $Y$ occur at a rate proportional to $XY$.

The SEIR model divides a population into susceptible, exposed, infectious, and recovered compartments.

If every transition transfers individuals between compartments without births, deaths, immigration, or emigration, the sum of all compartment sizes is constant.

Over a field containing the relevant roots of unity, $X^n-a$ is irreducible when the valuation data prevent $a$ from being a proper prime-divisor power, with the standard fourth-power exception.

The irreducible polynomials over $\mathbb{R}$ have degree one, or degree two with negative discriminant.

For an ideal $I$ of a commutative ring $R$, reducing coefficients modulo $I$ is a surjection $R[X]\to (R/I)[X]$ with kernel $I[X]$. Hence

A Boolean ring satisfies $r^2=r$ for every element. Expanding $(r+1{)}^2=r+1$ shows that it has characteristic two.

An ideal $\mathfrak{p}$ is prime when $ab\in \mathfrak{p}$ implies $a\in \mathfrak{p}$ or $b\in \mathfrak{p}$, equivalently when the quotient is an integral domain.

Module theory studies abelian groups with scalar multiplication by a ring. It extends linear algebra: a module over a field is a vector space, while ring scalars let the same language describe integer arithmetic and linear operators. The new phenomena are visible before any abstract classification: some nonzero scalars cannot be divided out, some modules have torsion elements of a module, and some have no basis of a module.

The distinction is useful because an integer relation such as $6m=0$ carries information which disappears if one assumes all nonzero scalars have multiplicative inverses. A finitely generated module over a principal ideal domain can be separated into a finite free module and a torsion module; the structure theorem for finitely generated modules over a principal ideal domain unifies the classification of finitely generated abelian groups with the study of linear operators through polynomial rings. See Keith Conrad's Modules over a PID, Theorem 4.1 and the operator applications.

A historical anchor is Emmy Noether's 1921 paper ^[ref]. Its first section studies rings and ideals under the condition that every ideal is a finitely generated module over its ring, and connects this to the ascending chain condition; later sections develop primary decomposition. This is an early source for the structural viewpoint underlying Noetherian rings. ^[ref] makes those statements accessible in English.

Every zero and pole of a square in $K(Y)$ has even order. Thus a rational function with a zero or pole of odd order cannot be a square. For characteristic different from two, this proves that $1-Y^2$ is not a square in $K(Y)$.

For a primitive polynomialif a prime $p$ does not divide $a_n$, divides every $a_j$ for $j<n$, and has $p^2\nmid a_0$, then $F$ is irreducible. If $F=GH$, reduction modulo $p$ forces both $\overline{G}$ and $\overline{H}$ to be positive-degree monomials. Their constant terms are consequently divisible by $p$, making $p^2\mid F(0)$, a contradiction. Gauss lemma for polynomials transfers the result between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$.

The content of a polynomial is a greatest common divisor of its coefficients, up to a unit.

Reduction between quotient rings sends units to units and is surjective when every target unit avoids the same prime factors.

Modulo p^e, a class is nilpotent exactly when its reduction modulo p is zero.

Modulo a prime power, a class is a unit exactly when it is not divisible by that prime.

A fiber product of rings consists of pairs having the same image in a third ring.

A commutative ring $R$ has a nontrivial idempotent $e$ exactly when it decomposes as a product of two nontrivial rings. The isomorphism determined by $e$ iswith inverse $(x,y)\mapsto x+y$.