Matches in DBpedia 2014 for { <http://dbpedia.org/resource/Zero_object_(algebra)> ?p ?o. }
Showing items 1 to 25 of
25
with 100 items per page.
- Zero_object_(algebra) abstract "In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and also has a trivial structure of abelian group. Aforementioned group structure usually identified as the addition, and the only element is called zero 0, so the object itself is denoted as {0}. One often refers to the trivial object (of a specified category) since every trivial object is isomorphic to any other (under a unique isomorphism).Instances of the zero object include, but are not limited to the following: As a group, the trivial group. As a ring, the trivial ring. As a module (over a ring R), the zero module. The term trivial module is also used, although it is ambiguous. As a vector space (over a field R), the zero vector space, zero-dimensional vector space or just zero space. As an algebra over a field or algebra over a ring, the trivial algebra.These objects are described jointly not only based on the common singleton and trivial group structure, but also because of shared category-theoretical properties.In the last three cases the scalar multiplication by an element of the base ring (or field) is defined as: κ0 = 0 , where κ ∈ R.The most general of them, the zero module, is a finitely-generated module with an empty generating set.For structures requiring the multiplication structure inside the zero object, such as the trivial ring, there is only one possible, 0 × 0 = 0, because there are no non-zero elements. This structure is associative and commutative. A ring R which has both an additive and multiplicative identity is trivial if and only if 1 = 0, since this equality implies that for all r within R, In this case it is possible to define division by zero, since the single element is its own multiplicative inverse. Some properties of {0} depend on exact definition of the multiplicative identity; see the section Unital structures below.Any trivial algebra is also a trivial ring. A trivial algebra over a field is simultaneously a zero vector space considered below. Over a commutative ring, a trivial algebra is simultaneously a zero module.The trivial ring is an example of a pseudo-ring of square zero. A trivial algebra is an example of a zero algebra.The zero-dimensional vector space is an especially ubiquitous example of a zero object, a vector space over a field with an empty basis. It therefore has dimension zero. It is also a trivial group over addition, and a trivial module mentioned above.".
- Zero_object_(algebra) thumbnail Terminal_and_initial_object.svg?width=300.
- Zero_object_(algebra) wikiPageExternalLink books?id=Nmg4AAAAIAAJ&pg=PA10.
- Zero_object_(algebra) wikiPageID "7455889".
- Zero_object_(algebra) wikiPageRevisionID "588678434".
- Zero_object_(algebra) author "Barile, Margherita".
- Zero_object_(algebra) author Margherita_Barile.
- Zero_object_(algebra) hasPhotoCollection Zero_object_(algebra).
- Zero_object_(algebra) id "TrivialModule".
- Zero_object_(algebra) id "ZeroModule".
- Zero_object_(algebra) title "Trivial Module".
- Zero_object_(algebra) title "Zero Module".
- Zero_object_(algebra) subject Category:Linear_algebra.
- Zero_object_(algebra) subject Category:Objects_(category_theory).
- Zero_object_(algebra) subject Category:Ring_theory.
- Zero_object_(algebra) subject Category:Zero.
- Zero_object_(algebra) comment "In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and also has a trivial structure of abelian group. Aforementioned group structure usually identified as the addition, and the only element is called zero 0, so the object itself is denoted as {0}.".
- Zero_object_(algebra) label "Zero object (algebra)".
- Zero_object_(algebra) label "Тривиальные объекты в алгебре".
- Zero_object_(algebra) sameAs m.09wtlw.
- Zero_object_(algebra) sameAs Q16089841.
- Zero_object_(algebra) sameAs Q16089841.
- Zero_object_(algebra) wasDerivedFrom Zero_object_(algebra)?oldid=588678434.
- Zero_object_(algebra) depiction Terminal_and_initial_object.svg.
- Zero_object_(algebra) isPrimaryTopicOf Zero_object_(algebra).
