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For a two-sided ideal $I$ of a ring $R$, the powers $I^n$ form a neighborhood base at zero for the adic topology. For an $R$-module $M$, use $I^{n}M$ on the left or $M{I}^n$ on the right. Convergence means that for every $n$ the differences eventually belong to the corresponding submodule. The formal power series ring $k[[t]]$ and formal power series module $M[[t]]$ are complete for $I=(t)$; convergence fixes each finite list of coefficients eventually. This is formal convergence, without an analytic condition on the size of coefficients.

For a field $k$, a homomorphism of $k$-algebras is a unital ring homomorphism preserving scalar multiplication by every element of $k$. For unital algebras it is equivalently a ring homomorphism commuting with their structural maps from $k$. Pullback by a morphism of algebraic varieties gives such a homomorphism between their coordinate rings.

An algebraic number is an algebraic integer when it satisfies a monic polynomial with integer coefficients. Algebraic integers are closed under addition and multiplication.

An algebra over a commutative ring $R$ is a ring $A$ with a specified unital ring homomorphism $R\to A$ whose image commutes with every element of $A$. Scalar multiplication is $r\cdot a=\phi (r)a$. For commutative $A$, the commutation requirement is automatic.

Suppose a ring has an underlying real normed vector space and its norm is multiplicative, without initially assuming real bilinearity of multiplication. For fixed $x$, distributivity makes $y\mapsto xy$ additive, while $\Vert xy-xz\Vert =\Vert x\Vert \Vert y-z\Vert$ makes it continuous. Additivity gives rational scalar compatibility, and continuity extends it to every real scalar. Apply the same argument in the other variable to obtain a bilinear map. The norm identity also excludes nonzero zero divisors.

An automorphism is an isomorphism from an object to itself.

The category of unital rings has identity-preserving ring homomorphisms as morphisms. The zero ring, allowing $0=1$, is terminal, and $\mathbb{Z}$ is initial. Nonzero rings admit no maps from the zero ring. Monomorphisms are injective: maps from $\mathbb{Z}[x]$ probe individual elements. It is a regular category, with surjective homomorphisms as regular epimorphisms; these are distinct from arbitrary categorical epimorphisms.

Changing the coefficient ring of a module along a ring homomorphism can mean restriction of scalars, extension of scalars, or coextension of scalars. The underlying module may be retained with a smaller ring action, or replaced by a tensor or Hom construction with the new ring.

The characteristic of a ring $R$ is the least positive integer $n$ satisfying $n\cdot 1_R=0$, or zero if no such integer exists. A nonzero ring of positive characteristic has prime characteristic when it is an integral domain.

A commutative ring is a ring whose multiplication satisfies $ab=ba$.

A family of ideals satisfies the descending chain condition when every descending sequence of ideals eventually becomes constant. A ring is Artinian precisely when all its ideals satisfy this condition.

The direct product of a family of rings consists of all tuples of their elements, with coordinatewise addition and multiplication. Unlike a direct sum, tuples may have infinitely many nonzero coordinates.

For a reduced ring with a nonempty disconnected spectrum of a commutative ring, choose a separation $V(I)\bigsqcup V(J)$. Disjointness gives $I+J=R$; covering the spectrum gives $IJ\subseteq \sqrt{(0)}=0$. For comaximal ideals, $I\cap J=IJ$. The Chinese remainder theorem then supplies a product of two nonzero quotient rings. Equivalently, if $i+j=1$ with $i\in I$, $j\in J$, then $i,j$ are nontrivial orthogonal idempotents. This connects topology of the spectrum to ring product decomposition by an idempotent without a Noetherian assumption.

A division ring is a nonzero ring in which every nonzero element has a multiplicative inverse. Its multiplication need not be commutative.

An endomorphism is a morphism from an object to itself.

Every proper ideal of a unital commutative ring is contained in a minimal prime over that ideal. A maximal ideal supplies a prime containing it, and an intersection of a decreasing chain of primes is still prime. Zorn's lemma, with reverse inclusion, therefore supplies a minimal member. No Noetherian assumption is needed.