Glossary

Ring theory has a vocabulary problem: semisimple, semiprimitive, semiprime, semilocal, semiperfect and semiprimary are six different conditions, and two of them have swapped meanings historically. This page defines every term used across the collection, with hypotheses in full.

Entries are grouped by what they describe rather than alphabetically, because in this subject the useful question is almost never what does this word mean but what does it mean relative to the neighbouring conditions.

The vocabulary is organised around one question: how far is a ring from being semisimple? Each term names a stage in the answer. Semilocal says the quotient by the radical is semisimple. Semiperfect adds that idempotents lift. Perfect adds T-nilpotency. Semiprimary adds nilpotence. Artinian adds a chain condition, and semisimple is the end of the line.

A second family — prime, semiprime, primitive, subdirectly irreducible — measures a different thing: how far a ring is from being a direct product of simpler pieces. The two families interlock but are not comparable, and confusing a member of one for a member of the other is the commonest terminological error in the subject.

Use with Notation Reference for the symbols and Ring Class Hierarchy for the containments between the classes named here.

Ring classes

Simple ring

$R\ne 0$ with no two-sided ideals except $0$ and $R$. Simple rings are left and right primitive $(11.6)$ but need not be artinian — the Weyl algebra $A_1(k)$ in characteristic $0$ is simple and not artinian.

Semiprime ring

${𝔄}^2=0$ forces $𝔄=0$ for ideals; equivalently ${\mathrm{Nil}}_{\ast }R=0$; equivalently $aRa=0$ forces $a=0$.

Reduced ring

No nonzero nilpotent elements. Reduced implies semiprime; a reduced ring is a subdirect product of domains $(12.7)$.

Subdirectly irreducible ring

$R\ne 0$ and the intersection of all nonzero ideals is nonzero $(12.2)$. Every nonzero ring is a subdirect product of subdirectly irreducible rings $(12.3)$.

Semilocal ring

$R/\mathrm{rad}R$ is semisimple. A commutative ring is semilocal exactly when it has finitely many maximal ideals.

Semiperfect ring

Semilocal, with idempotents lifting modulo $\mathrm{rad}R$ $(23.1)$; equivalently $1$ decomposes as a sum of orthogonal local idempotents $(23.6)$.

Right perfect ring

$R/\mathrm{rad}R$ is semisimple and $\mathrm{rad}R$ is right T-nilpotent $(23.18)$. Equivalently $R$ satisfies DCC on principal left ideals $(23.20)$ — the switch of sides is genuine.

Semiprimary ring

$\mathrm{rad}R$ is nilpotent and $R/\mathrm{rad}R$ is semisimple. Semiprimary implies both left and right perfect $(23.19)$; one-sided artinian implies semiprimary.

Von Neumann regular ring

For every $a$ there is $x$ with $axa=a$ $(4.23)$. Regular implies semiprimitive; regular plus left noetherian gives semisimple $(4.25)$.

Dedekind-finite ring

$xy=1$ implies $yx=1$. Fails in ${\mathrm{End}}_k(V)$ for ${\dim }_{k}V$ infinite.

Indecomposable ring

$R\ne 0$ with no central idempotents besides $0$ and $1$; equivalently $R$ is not a nontrivial direct product of rings $(22.4)$.

Basic ring

A semiperfect ring $R$ for which $R/\mathrm{rad}R$ is a finite direct product of division rings $(25.8)$. Every semiperfect ring is Morita equivalent to a basic one.

Division ring

$R\ne 0$ and every nonzero element is a unit. Commutative division rings are fields. A division ring is centrally finite if ${\dim }_{Z(R)}R$ is finite.

Ideals and radicals

Prime ideal

An ideal $𝔭⊊R$ with $𝔄𝔅\subseteq 𝔭$ forcing $𝔄\subseteq 𝔭$ or $𝔅\subseteq 𝔭$; equivalently $R/𝔭$ is a prime ring $(10.2)$.

Semiprime ideal

${𝔄}^2\subseteq 𝔮$ forces $𝔄\subseteq 𝔮$; equivalently an intersection of prime ideals $(10.11)$.

m-system

A set $S$ such that for $a,b\in S$ there is $r\in R$ with $arb\in S$. Complements of prime ideals are exactly the m-systems $(10.3)$.

n-system

A set $S$ such that for $a\in S$ there is $r$ with $ara\in S$; the multiplicative analogue for semiprime ideals $(10.10)$.

Lower nilradical

${\mathrm{Nil}}_{\ast }R=\sqrt{(0)}$, the intersection of all prime ideals; also called the Baer or prime radical $(10.13)$.

Upper nilradical

${\mathrm{Nil}}^{\ast }R$, the sum of all nil ideals; itself a nil ideal $(10.26)$.

Levitzki radical

$L\text{-}\mathrm{rad}R$, the largest locally nilpotent ideal; ${\mathrm{Nil}}_{\ast }R\subseteq L\text{-}\mathrm{rad}R\subseteq {\mathrm{Nil}}^{\ast }R\subseteq \mathrm{rad}R$ $(10.32)$.

Locally nilpotent set

Every finite subset generates a nilpotent subring without identity. Nilpotent implies locally nilpotent implies nil, both strictly.

T-nilpotent

$A$ is left T-nilpotent if every sequence $a_1,a_2,\dots$ from $A$ has $a_1{a}_2\cdots a_n=0$ for some $n$; right T-nilpotent reverses the order $(23.13)$. Nilpotent implies T-nilpotent implies nil.

Primitive ideal

An ideal $𝔄$ with $R/𝔄$ left primitive $(11.3)$; equivalently the annihilator of a simple left module $(11.4)$. $\mathrm{rad}R$ is their intersection $(11.5)$.

Augmentation ideal

The kernel of $\epsilon :kG\to k$, $\sum a_{g}g\mapsto \sum a_g$; free as a $k$-module on $\{g-1:g\ne 1\}$.

Small submodule

$N\subseteq M$ is small (or superfluous) if $N+L=M$ forces $L=M$ $(24.1)$. $\mathrm{rad}M$ is the sum of all small submodules $(24.4)$.

Modules

Faithful module

$\mathrm{ann}(M)=0$.

Indecomposable module

Nonzero, and not a direct sum of two nonzero submodules.

Strongly indecomposable module

$\mathrm{End}(M)$ is a local ring. Strongly indecomposable implies indecomposable; the converse needs a finiteness hypothesis $(19.17)$.

Composition series

A finite chain with simple factors. A module has one iff it is both noetherian and artinian $(1.19)$; the factors are unique up to permutation by Jordan–Hölder.

Socle

$\mathrm{soc}(M)$, the sum of the simple submodules, equivalently the largest semisimple submodule.

Radical of a module

$\mathrm{rad}M$, the intersection of the maximal submodules $(24.3)$. For $M=R$ this recovers $\mathrm{rad}R$; for a general $M$ it can be all of $M$.

Projective module

A direct summand of a free module $(2.7)$; equivalently every epimorphism onto it splits.

Flat module

$M{\otimes }_R-$ is exact $(24.20)$. Projective implies flat; the converse holds exactly over right perfect rings $(24.25)$.

Projective cover

An epimorphism $P\to M$ with $P$ projective and small kernel $(24.9)$. Unique up to isomorphism when it exists $(24.10)$.

Principal indecomposable module

A direct summand $eR$ of $R_R$ with $e$ a primitive idempotent $(25.1)$; over a semiperfect ring these are the projective covers of the simple modules.

Idempotents and decompositions

Primitive idempotent

$e\ne 0$ that is not the sum of two nonzero orthogonal idempotents $(21.15)$.

Local idempotent

$e\ne 0$ with $eRe$ a local ring. Local implies primitive; over a semiperfect ring the two coincide $(23.5)$.

Full idempotent

$e$ with $ReR=R$. For such $e$, $R$ and $eRe$ are Morita equivalent $(21.14)$.

Corner ring

$eRe$ for an idempotent $e$; it is a ring with identity $e$, and $\mathrm{End}(eR)\cong eRe$ $(21.7)$.

Peirce decomposition

$R=eRe\oplus eR(1-e)\oplus (1-e)Re\oplus (1-e)R(1-e)$ for an idempotent $e$ $(21.1)$.

Lifting idempotents

Given an ideal $I$ and $\overline{a}$ idempotent in $R/I$, finding an idempotent $e\in R$ with $e+I=\overline{a}$. Always possible when $I$ is nil $(21.28)$.

Block

An indecomposable direct factor of $R$; equivalently $R{e}_i$ for a primitive central idempotent $e_i$ in the decomposition $1=e_1+\cdots +e_n$ $(22.3)$.

Basic idempotent

An idempotent $e$ such that $eRe$ is a basic ring Morita equivalent to $R$; obtained by taking one idempotent from each isomorphism class of principal indecomposables $(25.5)$.

Group rings, representations and division algebras

Group ring

$kG=\underset{g\in G}{⨁}kg$ with multiplication induced from $G$ $(1.4)$.

Trivial unit

$ag$ with $a\in U(k)$, $g\in G$. Every group ring has these; whether it has others is Problem U $(6.16)$.

$p\prime$-group

A group with no element of order $p$. For finite groups this means $p\nmid \left|G\right|$.

FC group

A group all of whose elements have finitely many conjugates. $\Delta (G)$ denotes the subgroup of such elements $(6.23)$; a torsion-free FC group is abelian $(6.24)$.

Splitting field

A field $k$ for a $k$-algebra $A$ such that every simple $A$-module is absolutely irreducible, that is, remains simple under every field extension $(7.4)$.

Absolutely irreducible module

A simple module $V$ with $\mathrm{End}(V)=k$; equivalently $V{\otimes }_{k}K$ is simple for every extension $K/k$.

Ordered group

A group with a total order satisfying $g<h\Rightarrow agb<ahb$. For such $G$ and a domain $k$, $kG$ is a domain with only trivial units $(6.29)$.

Cyclic algebra

$(K/F,\sigma ,a)$: built from a cyclic Galois extension $K/F$ with group generated by $\sigma$ and an element $a\in F^{\times }$ $(14.5)$.

Formally real ring

A ring in which a sum of squares vanishes only when every term does; the algebraic prerequisite for an ordering $(17.11)$.

Positive cone

The set of non-negative elements of an ordered ring; closed under addition and multiplication and containing all squares $(17.1)$.

There is no standards body for algebraic terminology; ISO 80000-2 governs symbols and typography only. In practice the convention is set by the textbook a community learned from, which is why the same ring can be called semisimple in one paper and semiprimitive in the next.

• Semiprime and semiprimitive differ: ${\mathrm{Nil}}_{\ast }R=0$ versus $\mathrm{rad}R=0$. The second implies the first, never the reverse.
• Perfect always carries a side; semiperfect never does. The asymmetry is a theorem, not an oversight.
• Local ring here does not include a noetherian hypothesis, and does not require commutativity.
• Simple ring and simple module are unrelated conditions; a simple ring can have many non-isomorphic simple modules only if it is not artinian.

• Define semisimple explicitly the first time you use it in any document, however standard you believe your convention to be.
• Attach a side to every occurrence of artinian, noetherian, primitive and perfect.
• When introducing a radical, say which one and give its defining property, not just its symbol.
• Prefer the phrase zero Jacobson radical to any adjective when the audience is mixed.
• Cite the numbered result that pins a definition down — $(4.14)$ for semisimple, $(23.18)$ for perfect, $(19.1)$ for local — rather than relying on the word.

• 1893–1908Molien, Cartan, WedderburnAlgebras are classified; the radical is the largest nilpotent ideal and 'semisimple' names the quotient. The word already has its modern shape in the finite-dimensional case.
• 1927–29Artin and NoetherChain conditions replace finite dimension; 'completely reducible' becomes the standard name for what is now called semisimple, and module language takes over.
• 1945JacobsonThe radical is defined for arbitrary rings. Jacobson keeps 'semisimple' for zero radical, which is consistent with his own theory but conflicts with the older artinian usage.
• 1950s–60sThe radicals multiplyBaer, Levitzki and Brown-McCoy radicals arrive, along with the upper and lower nilradical vocabulary and the star notation.
• 1960BassPerfect and semiperfect rings are named; T-nilpotency enters the language, together with the deliberate side-switch in the definition of right perfect.
• 1970s onwardResolution by textbookAnderson-Fuller and later Lam adopt semiprimitive for zero radical, restoring 'semisimple' to its artinian meaning. Both usages survive in the literature and neither is going away.

Terminology in this subject records its history rather than its logic. Each name was coined for the most general setting available at the time, and later generalisations left the old word attached to a narrower class. Reading a definition is therefore also reading a date.