Polynomial ringIn mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers.
Ring (mathematics)In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.
Noetherian ringIn mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence of left (or right) ideals has a largest element; that is, there exists an n such that: Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated.
Category of groupsIn mathematics, the Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a . The study of this category is known as group theory. There are two forgetful functors from Grp, M: Grp → Mon from groups to monoids and U: Grp → Set from groups to . M has two adjoints: one right, I: Mon→Grp, and one left, K: Mon→Grp. I: Mon→Grp is the functor sending every monoid to the submonoid of invertible elements and K: Mon→Grp the functor sending every monoid to the Grothendieck group of that monoid.
Variety (universal algebra)In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras, and (direct) products.
Temperateness (virology)In virology, temperate refers to the ability of some bacteriophages (notably coliphage λ) to display a lysogenic life cycle. Many (but not all) temperate phages can integrate their genomes into their host bacterium's chromosome, together becoming a lysogen as the phage genome becomes a prophage. A temperate phage is also able to undergo a productive, typically lytic life cycle, where the prophage is expressed, replicates the phage genome, and produces phage progeny, which then leave the bacterium.
School libraryA school library (or a school library media center) is a library within a school where students, staff, and often, parents of a public or private school have access to a variety of resources. The goal of the school library media center is to ensure that all members of the school community have equitable access "to books and reading, to information, and to information technology." A school library media center "uses all types of media... is automated, and utilizes the Internet [as well as books] for information gathering.
Law libraryA law library is a special library used by law students, lawyers, judges and their law clerks, historians and other scholars of legal history in order to research the law. Law libraries are also used by people who draft or advocate for new laws, e.g. legislators and others who work in state government, local government, and legislative counsel offices or the U.S. Office of Law Revision Counsel and lobbying professionals. Self-represented, or pro se, litigants (parties to a civil lawsuit or criminal defendants who do not have a licensed attorney representing them) also use law libraries.
