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An integral domain is a nonzero commutative ring with identity and no zero divisors. A ring is Noetherian when every ideal is finitely generated, equivalently every ascending ideal chain stabilizes. A principal ideal domain is an integral domain in which every ideal is generated by one element.

Use the module criterion: an element $x$ is an algebraic integer if and only if it lies in a nonzero finitely generated $\mathbb{Z}$-module $M$ with $xM\subseteq M$.

Let the coefficients of the monic polynomial be $c_0,\dots ,c_{n-1}$ and put $A=\mathbb{Z}[c_0,\dots ,c_{n-1}]$. A ring generated by finitely many algebraic integers is a finitely generated $\mathbb{Z}$-module. If $\alpha$ is a root, thenis finitely generated, and the monic relationshows that $\alpha M\subseteq M$. The criterion proves that every root is an algebraic integer.

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 $N$.

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.

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

Since $\beta$ is integral, $R[\beta ]$ is generated over $R$ by finitely many powers, hence is fractional. It is invertible by (c). Because it is a ring, $I^2=I$; multiplying by $I^{-1}$ gives $I=R$. Thus every $\beta \in {\mathcal{O}}_K$ lies in $R$, so $R={\mathcal{O}}_K$.

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.

The second Adams operation $\chi \mapsto [g\mapsto \chi (g^2)]$ lies in the integral character ring, so $\tilde{\chi }=\sum n_i{\chi }_i$ with integers $n_i$. Norm one forces exactly one coefficient to be $\pm 1$; evaluation at the identity makes the sign positive. Squaring is bijective on conjugacy classes as well, so distinct irreducibles remain distinct. Thus the ${\tilde{\chi }}_i$ permute the irreducible characters.

A polynomial is a finite sum of monomials with coefficients in a ring. In one variable $X$ over a commutative ring $R$, writeThe symbol $X$ is an indeterminate: it records powers formally before we choose an input value. Finite expressions are easy to add, multiply, evaluate, and differentiate, making polynomials a common language for exact algebra and approximate numerical analysis.

Consider the real polynomialThe expanded form displays the coefficients and the degree of a polynomial; the factored form displays its roots of a polynomial. At $X=3$, either description gives $p(3)=4$. At $X=-1$ and $X=2$, the factored form makes the zero values immediate. Keeping several descriptions of one object is useful because each makes a different question easy.

The graph shows another connection. The factor $X+1$ changes sign across $-1$, while $(X-2{)}^2$ stays nonnegative across $2$. Thus the graph crosses the axis at the simple root of a polynomial and touches it at the double root. Multiplicity of a root turns an algebraic repetition into visible local behavior.

For calculations, Horner's method evaluates a polynomial through nested multiplication. For modeling, a Taylor polynomial describes local behavior near a point, while polynomial interpolation constructs a polynomial matching prescribed data. Their usefulness comes with assumptions: a local approximation and an exact identity are different claims, and changing the coefficient ring can change which polynomial roots exist.

A polynomial function is the function obtained by evaluating a polynomial on a specified ring or field. Distinct polynomials can induce the same polynomial function over a finite field; for example, $x^p$ and $x$ agree on ${\mathbb{F}}_p$.

For a prime ideal $P$ of a polynomial ring over a field, the quotient's Krull dimension is the transcendence degree of its fraction field, and its height is $n$ minus that degree. This dimension theorem identifies algebraic codimension with the length of prime chains. It must not be applied indiscriminately to every Noetherian ring without its polynomial-ring hypotheses.

A prime filtration of a module over a commutative ring is a finite increasing chain from zero to the module with each nonzero successive quotient module isomorphic to $R/P$ for a prime ideal $P$. Every finitely generated module over a Noetherian ring has one. In each nonzero quotient, a maximal annihilator of a module element is prime supplies a cyclic submodule $R/P$; lift it and repeat. The ascending chain condition for a Noetherian module makes the process terminate. This is not necessarily a composition series: the prime ideals need not be maximal.

The prime radical is the intersection of all prime ideals of a noncommutative ring. For a nonzero ring satisfying the ascending chain condition on two-sided ideals, there exist prime ideals of a noncommutative ring $P_1,\dots ,P_r$ with $P_1\cdots P_r=0$. Indeed, a maximal counterexample among two-sided ideals could not itself be a prime ideal of a noncommutative ring, and two larger witness ideals would contradict its maximality. Every prime ideal of a noncommutative ring contains one $P_i$, so the intersection equals ${\bigcap }_i{P}_i$, and its $r$th power is zero. This nilpotence assertion fails without a chain condition.

A nonzero central idempotent is primitive central if it is not a sum of two nonzero orthogonal central idempotents. For an Artinian ring, it determines a block of an Artinian algebra. This differs from a primitive idempotent in the whole ring: for a matrix algebra of size greater than one over a field, $1$ is primitive central but is a sum of diagonal idempotents.

For a two-sided ideal $I$ of a ring $R$, the quotient $R/I$ has addition and multiplication induced from $R$.

A ring is right Artinian if its right ideals satisfy the descending chain condition, equivalently its right regular module is an Artinian module. Left and right chain conditions should be distinguished for a general ring.

A multiplicative subset $S\subseteq A$ with $1\in S$ and $0\notin S$ of a ring $A$ satisfies the right Ore condition if for every $a\in A$ and $s\in S$ there are $u\in S$ and $b\in A$ such that $au=sb$. For a noncommutative domain, taking $S=A\setminus \{0\}$ gives a ring of right fractions $a{s}^{-1}$; every nonzero fraction is invertible. The usual additional denominator reversibility condition is automatic when the elements of $S$ are nonzero in a noncommutative domain.