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A module $P$ is projective when every map from $P$ through the target of a surjection lifts through that surjection.

A module $M$ over an integral domain $R$ is torsion-free when $rm=0$ with $0\ne r\in R$ implies $m=0$, equivalently when its torsion submodule is zero. Every free module over an integral domain is torsion-free, as one can check in each coefficient. Every vector space is torsion-free over its scalar field.

The converse requires hypotheses. A finitely generated module over a principal ideal domain is free if it is torsion-free, but the module $\mathbb{Q}$ over $\mathbb{Z}$ is torsion-free and not free. Any two nonzero rational numbers have a nontrivial integer relation, so a possible basis of a module could have at most one element; no single $q$ generates all rational numbers, since $q/2\notin \mathbb{Z}q$ for $q\ne 0$. The finite-generation hypothesis cannot be dropped. See Keith Conrad's Modules over a PID, Corollary 2.6.

A free module is a module with a basis of a module $(e_i{)}_{i\in I}$: every element has exactly one expression $m={\sum }_{i\in I}{r}_i{e}_i$ with $r_i\in R$ and only finitely many $r_i$ nonzero. Equivalently, the coefficient map ${⨁}_{i\in I}R\to M$ is a module isomorphism. Existence means generation; uniqueness means linear independence in a module. For finite $I$ of size $n$, this says $M\cong R^n$, a finite free module. See Neil Strickland's Rings, Modules and Linear Algebra, section 11.

The integer lattice ${\mathbb{Z}}^2$ in the opening diagram is a free module over $\mathbb{Z}$ with basis of a module $e_1=(1,0)$, $e_2=(0,1)$. For example, $(3,2)=3e_1+2e_2$, and comparing the two coordinates makes the coefficients unique. A different basis of a module is $u=(1,0)$, $v=(1,1)$, because $(a,b)=(a-b)u+bv$; in particular $(3,2)=u+2v$.

By contrast, $w_1=(2,0)$ and $w_2=(0,1)$ are linearly independent in a module, but generate only the submodule $2\mathbb{Z}\times \mathbb{Z}$. They miss $(1,0)$ because $2a=1$ has no integer solution. They would form a basis over the rational numbers. Even generation need not let one extract a basis of a module: the elements $2,3$ generate $\mathbb{Z}$, since $1=3-2$, but neither singleton generates it, and $3\cdot 2-2\cdot 3=0$ is a nontrivial linear relation between them. These are concrete failures of the usual vector space argument that divides by any nonzero coefficient.

The cyclic module $\mathbb{Z}/6\mathbb{Z}$ has no basis of a module over $\mathbb{Z}$: every element has the nontrivial relation $6m=0$, so no nonempty subset is linearly independent in a module, while the empty subset cannot generate this nonzero module. Specifying the scalar ring is essential: over $R=\mathbb{Z}/6\mathbb{Z}$, the same additive object is a free module with the one-element basis of a module $\{\overline{1}\}$, since each coefficient is itself a unique class in $R$.

For a submodule $N\le M$, the quotient module $M/N$ consists of additive cosets with scalar multiplication $r(m+N)=rm+N$.

A finitely generated module over a principal ideal domain decomposes aswith finitely many irreducibles $p$. The free rank and primary cyclic factors are unique up to associates and ordering.

The length of a module is the number of simple factors in a composition series. It is additive across short exact sequences.

A decreasing filtration of a module is a chain $M=M_0\supseteq M_1\supseteq \cdots$ of submodules.

An inverse system is a family $(N_i)$ with transition maps $g_{ij}:N_j\to N_i$ for $i\le j$, satisfying $g_{ii}=1$ and $g_{ik}=g_{ij}{g}_{jk}$.

A short exact sequence is an exact sequenceThus $j$ is injective, $q$ is surjective, and $\mathrm{im}j=\ker q$.

If a finitely generated faithful module $M$ over a commutative ring $R$ satisfies $xM\subseteq IM$, the determinant of a matrix presenting these relations gives a monic equationIn particular, an element of an extension field that preserves a nonzero finitely generated ideal is integral over $R$.

An $R$-module $M$ is flat when tensoring with it preserves injections, equivalently when $-{\otimes }_{R}M$ is an exact functor.

The tensor product of an $R$-module $M$ and an $R$-module $N$ is an $R$-module $M{\otimes }_{R}N$ generated by pure tensors $m\otimes n$, subject to additivity in both variables and the balancing relation

Laplace's integral method seeks $y(z)={\int }_C{e}^{zt}f(t)dt$. Substitution into the differential equation and integration by parts transfer multiplication by $z$ into differentiation with respect to $t$, producing a first-order equation for $f$ and endpoint conditions for $C$.

The Mertens function is the summatory Möbius function

Möbius inversion recovers an arithmetic function from its divisor sum. For arithmetic functions $f,g$, the following statements are equivalent, with each equality required for every positive integer $n$:Here $d$ ranges over positive integer divisors of $n$ and $\mu$ is the Möbius function. No multiplicativity assumption is needed, and both sums are finite. The right side can also be written ${\sum }_{d\mid n}\mu (n/d)g(d)$ by exchanging complementary integer divisors. ^[ref]

The divisor sum $g(n)$ bundles $f(n)$ together with values at smaller integer divisors. Möbius inversion subtracts aggregates indexed by one distinct prime factor, adds those indexed by two, and continues with the alternating signs of inclusion-exclusion. A divisor containing a squared prime factor receives weight zero. The Möbius divisor-sum identity ensures that every unwanted contribution cancels exactly.

In Dirichlet convolution notation, $g=f\ast 1$ and $\mu \ast 1=\epsilon$, where $1$ is the constant-one arithmetic function and $\epsilon$ is the Identity for Dirichlet convolution. Associativity and commutativity give the short proof:Convolving $f=\mu \ast g$ with $1$ proves the reverse implication. At $n=1$, the formula reduces to $f(1)=g(1)$ because $\mu (1)=1$.

For a worked example, take $f(n)=\phi (n)$, Euler's totient function. The totient divisor-sum identity says that $g(n)=n$, the identity arithmetic function:This identity comes from classifying the integers $1,\dots ,n$ by their greatest common divisor with $n$: the group with $\gcd (j,n)=h$ has $\phi (n/h)$ members, since $j/h$ and $n/h$ are coprime integers. Summing the group sizes gives $n$, as detailed under totient divisor-sum identity.

Now undo this aggregation with the Möbius function:The nonzero terms use $d=1,2,3,6$; $d=4,12$ vanish because they contain the square of the prime number $2$. The four surviving integers coprime to $12$ are $1,5,7,11$.

This example also reveals why Möbius inversion is useful: it turns an easy count of all objects into a count satisfying a coprimality restriction. Expanding over subsets of distinct prime factors givesThe product is inclusion-exclusion over multiples of the prime factors. For $12$, it gives $12(1-1/2)(1-1/3)=4$. More generally, whenever a counting problem produces a divisor sum, Möbius inversion lets that aggregate be reversed without solving a new system of equations.

A null curve has tangent $u$ satisfying $g(u,u)=0$. It represents the possible spacetime path of a light signal.

Its nonzero tangent is a null vector at every regular point; this distinguishes the curve from a single null direction.

The light cone of a spacetime event $O$ consists of $O$ together with the spacetime events having null separation from it. In the coordinates of an inertial frame centered on $O$, it obeysFor $t>0$ it traces an expanding spherical flash of light, with radius $ct$ at time $t$; for $t<0$ it describes possible incoming light rays. A spacetime diagram with one spatial coordinate shows the two lines $x=\pm ct$. The apparent $45^{\circ }$ angle uses equal scales for $x$ and $ct$ and is not a separate physical angle.

The light cone is the boundary, while its interior has timelike separation from $O$. An outgoing light ray follows the boundary; a future-directed massive observer's world line remains inside it. The exterior has spacelike separation: no signal travelling at or below the speed of light can connect its spacetime events directly to $O$. This separates possible causal influence from merely being assigned a later coordinate time.

In the diagram, $A=(4\mathrm{s},3\mathrm{ls})$ is inside the future boundary, $N=(3\mathrm{s},3\mathrm{ls})$ is on it, and $B=(2\mathrm{s},3\mathrm{ls})$ is outside. Their squared Minkowski intervals are respectively $-7,0,5$ in ${\mathrm{ls}}^2$. Thus $B$ occurs later than $O$ in the displayed inertial frame while still being unable to receive a light-speed-or-slower signal sent from $O$.

The usefulness is immediate for an experiment: a detector's reading can depend on a source event only if a causal influence can reach it. A Lorentz boost changes coordinates but preserves this classification. For a boost with $\beta =0.8$ and Lorentz factor $\gamma =5/3$, applying the Lorentz transformation gives, in light-second length units,The corresponding squared Minkowski intervals remain $-7,0,5$. In particular $B$ is now earlier than $O$, which is allowed for spacelike separation; $A$ remains later.

The Minkowski norm is the Lorentz-invariant quadratic form $U\cdot U={\eta }_{\mu \nu }{U}^{\mu }{U}^{\nu }$.

This quadratic form belongs to the geometry of Minkowski spacetime; the spacetime itself is a broader concept than this squared norm.

For a real quadratic field $K$ with embeddings ${\sigma }_1,{\sigma }_2$, the imageis a lattice in ${\mathbb{R}}^2$ of covolume $\sqrt{|d_K|}$, and the product of the two coordinates is the field norm.

For a dominated family with positive densities $p_{\theta }$, a statistic $T$ is minimal sufficient if