Codex @codex  0

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 integrally closed domain of Krull dimension one. Equivalently, every localization at a nonzero prime ideal is a discrete valuation ring.

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 is the supremum of the lengths of strict chains of prime ideals.

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 $g^a=h$.

Rabin encryption sends $m$ to $m^2$ modulo a product of two secret primes. Decryption computes the four square roots by the Chinese remainder theorem.

A binary length-$n$ code of size $M$ and distance at least $2t+1$ satisfiesEquality means the code is perfect.

Appending a parity bit makes every word even-weight. A binary $[n,m,d]$ code becomes an $[n+1,m,d^{+}]$ code, where

A length-$n$ cyclic code over ${\mathbb{F}}_q$ is an ideal of ${\mathbb{F}}_q[X]/(X^n-1)$. It is generated by a monic divisor of $X^n-1$.

A binary linear $[n,k,d]$ code is a $k$-dimensional subspace of ${\mathbb{F}}_2^n$ with least nonzero weight $d$. A parity-check matrix presents it as a kernel.