In the branch of mathematics called

category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, a hopfian object is an object ''A'' such that any

epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \ ...

of ''A'' onto ''A'' is necessarily an

automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...

. The dual notion is that of a cohopfian object, which is an object ''B'' such that every

monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism ...

from ''B'' into ''B'' is necessarily an automorphism. The two conditions have been studied in the categories of

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...

s, and

topological spaces In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

. The terms "hopfian" and "cohopfian" have arisen since the 1960s, and are said to be in honor of

Heinz Hopf Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry. Early life and education Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Elizabeth ( ...

and his use of the concept of the hopfian group in his work on

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

s of surfaces.

Properties

Both conditions may be viewed as types of finiteness conditions in their category. For example, assuming Zermelo–Fraenkel set theory with the axiom of choice and working in the

category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition of m ...

, the hopfian and cohopfian objects are precisely the

finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. Th ...

s. From this it is easy to see that all finite groups, finite modules and finite rings are hopfian and cohopfian in their categories. Hopfian objects and cohopfian objects have an elementary interaction with

projective object In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object. ...

s and

injective object In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. ...

s. The two results are: *An injective hopfian object is cohopfian. *A projective cohopfian object is hopfian. The proof for the first statement is short: Let ''A'' be an injective hopfian object, and let ''f'' be an injective morphism from ''A'' to ''A''. By injectivity, ''f'' factors through the identity map ''I''_''A'' on ''A'', yielding a morphism ''g'' such that ''gf''=''I''_''A''. As a result, ''g'' is a surjective morphism and hence an automorphism, and then ''f'' is necessarily the inverse automorphism to ''g''. This proof can be dualized to prove the second statement.

Hopfian and cohopfian groups

Hopfian and cohopfian modules

Here are several basic results in the category of modules. It is especially important to remember that ''R''_''R'' being hopfian or cohopfian as a module is different from ''R'' being hopfian or cohopfian as a ring. * A

Noetherian module In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion. Historically, Hilbert was the first mathematician to work with the propert ...

is hopfian, and an

Artinian module In mathematics, specifically abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it ...

is cohopfian. * The module ''R''_''R'' is hopfian if and only if ''R'' is a directly finite ring. Symmetrically, these two are also equivalent to the module _''R''''R'' being hopfian. * In contrast with the above, the modules ''R''_''R'' or _''R''''R'' can be cohopfian or not in any combination. An example of a ring cohopfian on one side but not the other side was given in . However, if either of these two modules is cohopfian, ''R'' is hopfian on both sides (since ''R'' is projective as a left or right module) and directly finite.

Hopfian and cohopfian rings

The situation in the category of rings is quite different from the category of modules. The morphisms in the category of rings with unity are required to preserve the identity, that is, to send 1 to 1. * If ''R'' satisfies the

ascending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These con ...

on ideals, then ''R'' is hopfian. This can be proven by analogy with the fact for Noetherian modules. The counterpart idea for "cohopfian" does not exist however, since if ''f'' is a ring homomorphism from ''R'' into ''R'' preserving identity, and the image of ''f'' is not ''R'', then the image is certainly not an ideal of ''R''. In any case, this shows that a one sided Noetherian or Artinian ring is always hopfian. * Any simple ring is hopfian, since the kernel of any endomorphism is an ideal, which is necessarily zero in a simple ring. In contrast, in , an example of a non-cohopfian

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

was given. * The

full linear ring In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero. ...

End_D(V) of a countable dimensional vector space is a hopfian ring which is not hopfian as a module, since it only has three ideals, but it is not directly finite. The paper also gives an example of a cohopfian ring which is not cohopfian as a module. * Also in , it is shown that for a

Boolean ring In mathematics, a Boolean ring ''R'' is a ring for which ''x''2 = ''x'' for all ''x'' in ''R'', that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean al ...

''R'' and its associated

Stone space In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact totally disconnected Hausdorff space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in t ...

''X'', the ring ''R'' is hopfian in the category of rings if and only if ''X'' is cohopfian in the category of topological spaces, and ''R'' is cohopfian as a ring if and only if ''X'' is hopfian as a topological space.

Hopfian and cohopfian topological spaces

* In , a series of results on compact manifolds are included. Firstly, the only

compact manifold In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example is ...

s which are hopfian are finite

discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...

s. Secondly, compact manifolds without boundary are always cohopfian. Lastly, compact manifolds with nonempty boundary are not cohopfian.

References

* * * *{{citation , last=Varadarajan , first=K. , contribution=Some recent results on Hopficity, co-Hopficity and related properties , title=International Symposium on Ring Theory , series=Trends Math. , publisher=Birkhäuser Boston , year=2001 , pages=371–392 , mr=1851216

External links

Hopfian groupCo-hopfian group
Category theory Group theory Module theory Ring theory