GalliumGallium is a chemical element with the symbol Ga and atomic number 31. Discovered by the French chemist Paul-Émile Lecoq de Boisbaudran in 1875, gallium is in group 13 of the periodic table and is similar to the other metals of the group (aluminium, indium, and thallium). Elemental gallium is a relatively soft, silvery metal at standard temperature and pressure. In its liquid state, it becomes silvery white. If enough force is applied, solid gallium may fracture conchoidally.
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.
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.
Artinian ringIn mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields.
Local ringIn mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal.
DiboraneDiborane(6), commonly known as diborane, is the chemical compound with the formula B2H6. It is a toxic, colorless, and pyrophoric gas with a repulsively sweet odor. Given its simple formula, borane is a fundamental boron compound. It has attracted wide attention for its electronic structure. Several of its derivatives are useful reagents. The structure of diborane has D2h symmetry. Four hydrides are terminal, while two bridge between the boron centers. The lengths of the B–Hbridge bonds and the B–Hterminal bonds are 1.
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.
GermaniumGermanium is a chemical element with the symbol Ge and atomic number 32. It is lustrous, hard-brittle, grayish-white and similar in appearance to silicon. It is a metalloid in the carbon group that is chemically similar to its group neighbors silicon and tin. Like silicon, germanium naturally reacts and forms complexes with oxygen in nature. Because it seldom appears in high concentration, germanium was discovered comparatively late in the discovery of the elements.
Aluminium hydrideAluminium hydride (also known as alane and alumane) is an inorganic compound with the formula AlH3. Alane and its derivatives are common reducing (hydride addition) reagents in organic synthesis that are used in solution at both laboratory and industrial scales. In solution—typically in etherial solvents such tetrahydrofuran or diethyl ether—aluminium hydride forms complexes with Lewis bases, and reacts selectively with particular organic functional groups (e.g.
CarbideIn chemistry, a carbide usually describes a compound composed of carbon and a metal. In metallurgy, carbiding or carburizing is the process for producing carbide coatings on a metal piece. The carbides of the group 4, 5 and 6 transition metals (with the exception of chromium) are often described as interstitial compounds. These carbides have metallic properties and are refractory. Some exhibit a range of stoichiometries, being a non-stoichiometric mixture of various carbides arising due to crystal defects.
