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

Given ring homomorphisms $f:A\to C$ and $g:B\to C$, their fiber product of rings isIt is a subring of the direct product of rings with coordinatewise arithmetic. A compatible pair of homomorphisms from any ring factors through it uniquely, its defining universal property.

For example, $\mathbb{Z}{\times }_{{\mathbb{F}}_2}\mathbb{Z}$ consists of integer pairs of the same parity. Unlike the full direct product, it glues the two coordinates along their common residue. This construction supports patching modules and expresses rings assembled from pieces with an agreed overlap.

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 $(x,y)\mapsto x+y$.

An integral domain is a nonzero commutative ring without zero divisors. Equivalently, a product of two nonzero elements is always nonzero. The zero ring is excluded, and $\mathbb{Z}/6\mathbb{Z}$ fails because $2\cdot 3=0$. Every field is an integral domain, but the integers show that the converse fails: $2$ has no multiplicative inverse in $\mathbb{Z}$. See MIT's Basic Properties of Rings, definitions 15.9–15.10 and example 15.11.

The practical benefit is multiplicative cancellation. In the integers, $2x=6$ has at most one solution and indeed $x=3$ works. Yet $2x=1$ has no integer solution. Thus multiplicative cancellation guarantees uniqueness when a solution exists; it does not guarantee division by every nonzero element. Passing to the fraction field $\mathbb{Q}$ supplies the missing solution $1/2$.

The polynomial ring $\mathbb{Z}[X]$ is also an integral domain. For example, the product of $2X+1$ and $3X-4$ has leading coefficient $6$ and cannot vanish. Generally the leading coefficient of a product is the product of the two leading coefficients; an integral domain prevents that coefficient from vanishing. This connects integer arithmetic to polynomial arithmetic through one reusable principle.

Historically, this connection grew far beyond elementary examples. Keith Conrad identifies the ring of integers of a number field and a ring of polynomial functions on a smooth plane curve as early important examples of Dedekind domains. This shared structure connects number theory with algebraic geometry; see Conrad's Ideal Factorization, section 6.

A ring is a set $R$ with two binary operations, addition and multiplication, satisfying three requirements. Under addition, $R$ is an abelian group: there is a zero, every element has an inverse element, and grouping or exchanging summands does not change the result. Multiplication is associative and has an identity element $1$, so $(ab)c=a(bc)$ and $1a=a1=a$. Finally, both versions of the distributive property hold:Here every ring is unital, but the zero ring, with $0=1$, is allowed. Commutativity of multiplication is an additional condition defining a commutative ring. These conventions agree with ^[ref].

The integers form a ring with their usual operations. The polynomial ring $\mathbb{Z}[X]$ is another: addition combines coefficients, while multiplication uses the distributive property. For example,The same rules justify symbolic calculations in both rings; this is why one definition can organize apparently different kinds of arithmetic. In every ring, $a0=0a=0$, and $a(-b)=-(ab)$. The first identity follows by distributing $a(0+0)$ and subtracting $a0$; the second follows by distributing $a(b+(-b))$. Thus familiar sign rules are consequences of the structure.

Order still matters in some rings. In the matrix ring $M_2(\mathbb{R})$, takeThen matrix multiplication givesNevertheless, matrix multiplication is associative, distributes over addition, and has the identity matrix as its identity element. This is a valid ring. On a column vector, $AB$ applies $B$ first and $A$ second: swapping coordinates before discarding the second coordinate differs from doing those actions in reverse. The failure of commutativity records a meaningful difference in function composition.

A Dedekind domain is a Noetherian ring that is an integrally closed domain of Krull dimension one. Equivalently, for a Noetherian domain that is not a field, every localization at a prime ideal at a nonzero prime is a discrete valuation ring. The Noetherian and nonfield qualifications belong to this local characterization; a field would satisfy its condition vacuously.

The ring of integers of a number field is a standard example. Every nonzero proper ideal has a unique factorization into nonzero prime ideals, even when element factorization is not unique. Locally, the exponent of each prime is the discrete valuation of that ideal in its corresponding discrete valuation ring. This connects unique factorization of ideals in a Dedekind domain with local arithmetic.

A local ring is a commutative ring with exactly one maximal ideal.

A proper ideal $\mathfrak{m}$ of a commutative ring $R$ is maximal when no proper ideal lies strictly between $\mathfrak{m}$ and $R$. Equivalently, $R/\mathfrak{m}$ is a field.

The polynomial ring $R[X]$ consists of polynomials in an indeterminate $X$ with coefficients in a ring $R$.

The Krull dimension of a commutative ring $R$ is the supremum of the lengths $n$ of strict chains of prime ideals ${\mathfrak{p}}_0⊊\cdots ⊊{\mathfrak{p}}_n$. The length counts inclusions, not the number of ideals.

A field has dimension zero. The integers have dimension one, with chains $(0)⊊(p)$, and $k[X_1,\dots ,X_n]$ over a field has dimension $n$. In algebraic geometry, chains of primes correspond to nested irreducible closed subsets, so this algebraic definition captures geometric dimension. It is distinct from the dimension of a vector space.

An element $a$ of an $R$-algebra is integral over $R$ when it satisfies a monic polynomial with coefficients in $R$.

For a multiplicatively closed set $S$, the localization $S^{-1}R$ formally makes every element of $S$ invertible. Elements are represented by fractions $r/s$.

Minimum-distance decoding chooses a codeword closest to the received word; on a binary symmetric channel with $p<1/2$, it is maximum-likelihood decoding under Hamming distance.

Maximum-likelihood decoding chooses the codeword under which the received word has greatest conditional probability.

Maximum a posteriori decoding chooses the codeword maximizing prior probability times channel likelihood and minimizes average decision error.

A binary symmetric channel is a discrete memoryless channel that flips each transmitted bit with fixed crossover probability $p$. For one use, write $Y=X\oplus Z$, where $\oplus$ is exclusive or and $Z$ is a Bernoulli random variable with parameter $p$, independent of the input $X$. Thus $Z=0$ preserves the bit and $Z=1$ flips it. Different uses have independent noise under this model.

With row labels $X=0,1$ and column labels $Y=0,1$, its transition matrix isEach row sums to one, and exchanging both binary labels leaves the rule unchanged, explaining symmetric. For $p=0.1$, a prescribed three-bit input $010$ produces $010$ with probability $0.9^3=0.729$ and $011$ with probability $0.9^2(0.1)=0.081$. The received $011$ does not itself announce the changed position: it could also arise from input $011$ with no flips.

The binary symmetric channel capacity quantifies how well a block code can overcome this uncertainty, and input distribution optimization for a binary symmetric channel shows why a fair input attains the best information rate. A completely noisy binary symmetric channel has $p=1/2$ and carries no input information; a known flip on every use, $p=1$, is instead reversible.

A code is decipherable, or uniquely decodable, when every finite concatenation of codewords has at most one decomposition into source codewords.

Alice publishes $g^a$ and Bob publishes $g^b$ in a public cyclic group; each can then compute the shared secret $g^{ab}$. Its security relies on the difficulty of recovering the exponents or otherwise solving the computational Diffie--Hellman problem.

Given a cyclic group generator $g$ and an element $h$, the discrete logarithm problem asks for an exponent $a$ such that $g^a=h$.

The Rabin cryptosystem uses modular squaring with public modulus $N=pq$, where $p$ and $q$ are distinct odd prime numbers kept secret. The raw encryption map sends a message representative $m$ toKnowing the prime factorization lets the receiver compute modular square roots modulo each prime and combine them by the Chinese remainder theorem. For a message coprime to $N$, its ciphertext has exactly four roots modulo $N$, corresponding to the two sign choices modulo each prime. A full encryption scheme therefore needs a specified message encoding or other rule to select the intended root; raw modular squaring by itself is four-to-one on the units.^[ref]

For example, $N=7\cdot 11=77$ and $m=10$ give $c=23$. Its four roots are $10,32,45,67$ modulo $77$. The relation to integer factorization is especially direct: two roots that differ by more than an overall sign reveal a nontrivial factor through a greatest common divisor. The Rabin-Williams scheme is a more specific variant, rather than another name for every Rabin cryptosystem.