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.
Ring homomorphismIn ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is a function f : R → S such that f is: addition preserving: for all a and b in R, multiplication preserving: for all a and b in R, and unit (multiplicative identity) preserving: Additive inverses and the additive identity are part of the structure too, but it is not necessary to require explicitly that they too are respected, because these conditions are consequences of the three conditions above.
Group ringIn algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.
Quotient ringIn ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring R and a two-sided ideal I in R, a new ring, the quotient ring R / I, is constructed, whose elements are the cosets of I in R subject to special + and ⋅ operations.
Escherichia coliEscherichia coli (ˌɛʃəˈrɪkiə_ˈkoʊlaɪ ) is a Gram-negative, facultative anaerobic, rod-shaped, coliform bacterium of the genus Escherichia that is commonly found in the lower intestine of warm-blooded organisms. Most E. coli strains are harmless, but some serotypes such as EPEC, and ETEC are pathogenic and can cause serious food poisoning in their hosts, and are occasionally responsible for food contamination incidents that prompt product recalls.
GroELGroEL is a protein which belongs to the chaperonin family of molecular chaperones, and is found in many bacteria. It is required for the proper folding of many proteins. To function properly, GroEL requires the lid-like cochaperonin protein complex GroES. In eukaryotes the organellar proteins Hsp60 and Hsp10 are structurally and functionally nearly identical to GroEL and GroES, respectively, due to their endosymbiotic origin. HSP60 is implicated in mitochondrial protein import and macromolecular assembly.
Ring of integersIn mathematics, the ring of integers of an algebraic number field is the ring of all algebraic integers contained in . An algebraic integer is a root of a monic polynomial with integer coefficients: . This ring is often denoted by or . Since any integer belongs to and is an integral element of , the ring is always a subring of . The ring of integers is the simplest possible ring of integers. Namely, where is the field of rational numbers. And indeed, in algebraic number theory the elements of are often called the "rational integers" because of this.
Macromolecular assemblyThe term macromolecular assembly (MA) refers to massive chemical structures such as viruses and non-biologic nanoparticles, cellular organelles and membranes and ribosomes, etc. that are complex mixtures of polypeptide, polynucleotide, polysaccharide or other polymeric macromolecules. They are generally of more than one of these types, and the mixtures are defined spatially (i.e., with regard to their chemical shape), and with regard to their underlying chemical composition and structure.
OligomerIn chemistry and biochemistry, an oligomer (əˈlɪgəmər) is a molecule that consists of a few repeating units which could be derived, actually or conceptually, from smaller molecules, monomers. The name is composed of Greek elements oligo-, "a few" and -mer, "parts". An adjective form is oligomeric. The oligomer concept is contrasted to that of a polymer, which is usually understood to have a large number of units, possibly thousands or millions. However, there is no sharp distinction between these two concepts.
Developmental disorderDevelopmental disorders comprise a group of psychiatric conditions originating in childhood that involve serious impairment in different areas. There are several ways of using this term. The most narrow concept is used in the category "Specific Disorders of Psychological Development" in the ICD-10. These disorders comprise developmental language disorder, learning disorders, motor disorders, and autism spectrum disorders. In broader definitions ADHD is included, and the term used is neurodevelopmental disorders.
