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.
Algebraic groupIn mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties.
Multiplicatively closed setIn abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset S of a ring R such that the following two conditions hold: for all . In other words, S is closed under taking finite products, including the empty product 1. Equivalently, a multiplicative set is a submonoid of the multiplicative monoid of a ring. Multiplicative sets are important especially in commutative algebra, where they are used to build localizations of commutative rings. A subset S of a ring R is called saturated if it is closed under taking divisors: i.
Algebraic closureIn mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemma or the weaker ultrafilter lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.
Clopen setIn topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open closed, and therefore clopen.
Algebraic operationIn mathematics, a basic algebraic operation is any one of the common operations of arithmetic, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power). These operations may be performed on numbers, in which case they are often called arithmetic operations. They may also be performed, in a similar way, on variables, algebraic expressions, and more generally, on elements of algebraic structures, such as groups and fields.
Upper setIn mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X) of a partially ordered set is a subset with the following property: if s is in S and if x in X is larger than s (that is, if ), then x is in S. In other words, this means that any x element of X that is to some element of S is necessarily also an element of S. The term lower set (also called a downward closed set, down set, decreasing set, initial segment, or semi-ideal) is defined similarly as being a subset S of X with the property that any element x of X that is to some element of S is necessarily also an element of S.
HangulThe Korean alphabet, known as Hangul (ˈhɑːnguːl ; ) in South Korea and Chosŏn'gŭl () in North Korea, is the modern official writing system for the Korean language. The letters for the five basic consonants reflect the shape of the speech organs used to pronounce them, and they are systematically modified to indicate phonetic features; similarly, the vowel letters are systematically modified for related sounds, making Hangul a featural writing system.
AntAnts are eusocial insects of the family Formicidae and, along with the related wasps and bees, belong to the order Hymenoptera. Ants evolved from vespoid wasp ancestors in the Cretaceous period. More than 13,800 of an estimated total of 22,000 species have been classified. They are easily identified by their geniculate (elbowed) antennae and the distinctive node-like structure that forms their slender waists.
Myrmecia (ant)Myrmecia is a genus of ants first established by Danish zoologist Johan Christian Fabricius in 1804. The genus is a member of the subfamily Myrmeciinae of the family Formicidae. Myrmecia is a large genus of ants, comprising at least 93 species that are found throughout Australia and its coastal islands, while a single species is only known from New Caledonia. One species has been introduced out of its natural distribution and was found in New Zealand in 1940, but the ant was last seen in 1981.
Kim Il SungKim Il Sung (ˈkɪm_ˈɪlˈsʌŋ,_-ˈsʊŋ; , kimils͈ʌŋ; 15 April 1912 – 8 July 1994) was a Korean politician and the founder of North Korea. He ruled the country from the country's establishment in 1948 until his death in 1994. Afterwards, he was declared its eternal president. His birth name was Kim Song Ju (). He held the posts of the Premier from 1948 to 1972 and President from 1972 to 1994. He was the leader of the Workers' Party of Korea (WPK) from 1949 to 1994 (titled as Chairman from 1949 to 1966 and as General Secretary after 1966).
Kim Jong IlKim Jong Il (,kɪm_ʤɒŋˈɪl; ; kim.dzɔŋ.il; also transcribed as Kim Jong-il and born Yuri Irsenovich Kim; 16 February 1941 or 1942 – 17 December 2011) was a North Korean politician who was the second supreme leader of North Korea from 1994 to 2011. He led North Korea from the 1994 death of his father Kim Il Sung, the first Supreme Leader, until his own death in 2011, when he was succeeded by his son, Kim Jong Un. In the early 1980s, Kim had become the heir apparent for the leadership of the Democratic People's Republic of Korea (DPRK) and assumed important posts in the party and army organs.
