Domain

In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0.That is, it is a ring with the zero-product property. Some authors requires the ring to have 1 ≠ 0,and sometimes just to be nontrivial. In other words, a domain is a nontrivial ring without left or right zero divisors. A commutative domain is called an integral domain.

A finite domain is automatically a finite field by Wedderburn's little theorem.

Zero-divisors have a geometric interpretation, at least in the case of cNo countexamples are known, but the problem remains open in general (as of 2007). For many special classes of groups, the answer is affirmative. Farkas and Snider proved in 1976 that if G is a torsion-free polycyclic-by-finite group and char K = 0 then the group ring K[G] is a domain. Later (1980) Cliff removed the restriction on the characteristic of the field. In 1988, Kropholler, Linnell and Moody generalized these results to the case of torsion-free solvable and solvable-by-finite groups. Earlier (1965) work of Lazard, whose importance was not appreciated by the specialists in the field for about 20 years, had dealt with the case where K is the ring of p-adic integers and G is the pth congruence subgroup of GL(n,Z).ommutative rings: a ring R is an integral domain, if and only if it is reduced and its spectrum Spec R is an irreducible topological space. The first property is often considered to encode some infinitesimal information, where the second one is of geometric nature.

An example: the ring k[x, y]/(xy), where k is a field, is not a domain, as the images of x and y in this ring are zero-divisors. Geometrically, this corresponds to the fact that the spectrum of this ring, which is the union of the lines x = 0 and y = 0, is not irreducible. Indeed, these two lines are its irreducible components.

Integral domain

In abstract algebra, an integral domain is a commutative ring with 1 ≠ 0 that has no zero divisors. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility. An integral domain is a commutative domain with identity.

Viewing the underlying commutative ring as a preadditive category, the above criterion on zero divisors is equivalent to the condition that every nonzero morphism is a monomorphism (hence also an epimorphism, by making use of the bilinear structure on the set of morphisms).

The condition 1 ≠ 0 only serves to exclude the trivial ring {0}.

The above is how "integral domain" is almost universally defined, but there is some variation. In particular, noncommutative integral domains are sometimes admitted, and very rarely the condition 1 ≠ 0 is omitted. However, we follow the much more usual convention of reserving the term integral domain for the commutative case and use domain for the noncommutative case. Some sources, notably Lang, use the term entire ring for integral domain

Domains

In mathematics, more specifically ring theory, an atomic domain is an integral domain, every non-zero non-unit of which, is a finite product of irreducible elements. Atomic domains are different from unique factorization domains, because this finite decomposition of an element into irreducibles need not be unique; equivalently, not every irreducible element is a prime. Important examples of atomic domains include the class of all unique factorization domains, and all Noetherian domains. In particular, any integral domain satisfying the ascending chain condition on principal ideals (i.e. the ACCP), is a atomic domain. Despite the claim that appears in Cohn's famous paper, the converse is known to be falseThe ring of integers (that is, the set of integers with the natural operations of addition and multiplication) satisfy many important properties. One such property is the fundamental theorem of arithmetic. Thus, when considering abstract rings, a natural question to ask is under what conditions such a theorem holds. Since a unique factorization domain is precisely a ring in which an analogue of the fundamental theorem of arithmetic holds, this question is readily answered. However, one notices that there are two aspects of the fundamental theorem of the arithmetic; that is, any integer is the finite product of prime numbers, as well as that this product is unique up to rearrangement (and multiplication by units). Therefore, it is also natural to ask under what conditions particular elements of a ring can be "decomposed" without requiring uniqueness.

Internet Protocol

The Internet Protocol (IP) is a protocol used for communicating data across a packet-switched internetwork using the Internet Protocol Suite, also referred to as TCP/IP.

IP is the primary protocol in the Internet Layer of the Internet Protocol Suite and has the task of delivering distinguished protocol datagrams (packets) from the source host to the destination host solely based on their addresses. For this purpose the Internet Protocol defines addressing methods and structures for datagram encapsulation. The first major version of addressing structure, now referred to as Internet Protocol Version 4 (IPv4) is still the dominant protocol of the Internet, although the successor, Internet Protocol Version 6 (IPv6) is being deployed actively worldwide.Data from an upper layer protocol is encapsulated as packets/datagrams (the terms are basically synonymous in IP). Circuit setup is not needed before a host may send packets to another host that it has previously not communicated with (a characteristic of packet-switched networks), thus IP is a connectionless protocol. This is in contrast to public switched telephone networks that require the setup of a circuit for each phone call Because of the abstraction provided by encapsulation, IP can be used over a heterogeneous network, i.e., a network connecting computers may consist of a combination of Ethernet, ATM, FDDI, Wi-Fi, token ring, or others. Each link layer implementation may have its own method of addressing (or possibly the complete lack of it), with a corresponding need to resolve IP addresses to data link addresses. This address resolution is handled by the Address Resolution Protocol (ARP) for IPv4 and Neighbor Discovery Protocol (NDP) for IPv6.

IP address

An Internet Protocol (IP) address is a numerical label that is assigned to devices participating in a computer network that uses the Internet Protocol for communication between its nodes. An IP address serves two principal functions: host or network interface identification and location addressing. Its role has been characterized as follows: "A name indicates what we seek. An address indicates where it is. A route indicates how to get there."

The designers of TCP/IP defined an IP address as a 32-bit number and this system, known as Internet Protocol Version 4 or IPv4, is still in use today. However, due to the enormous growth of the Internet and the resulting depletion of available addresses, a new addressing system (IPv6), using 128 bits for the address, was developed in 1995 and last standardized by RFC 2460 in 1998.^[4] Although IP addresses are stored as binary numbers, they are usually displayed in human-readable notations, such as 208.77.188.166 (for IPv4), and 2001:db8:0:1234:0:567:1:1 (for IPv6).

The Internet Protocol also routes data packets between networks; IP addresses specify the locations of the source and destination nodes in the topology of the routing system. For this purpose, some of the bits in an IP address are used to designate a subnetwork. The number of these bits is indicated in CIDR notation, appended to the IP address; e.g., 208.77.188.166/24.

As the development of private networks raised the threat of IPv4 address exhaustion, RFC 1918 set aside a group of private address spaces that may be used by anyone on private networks. They are often used with network address translators to connect to the global public Internet.

The Internet Assigned Numbers Authority (IANA), which manages the IP address space allocations globally, cooperates with five Regional Internet Registries (RIRs) to allocate IP address blocks to Local Internet Registries (Internet service providers) and other entities.

HOST NAME

A hostname is a label that is assigned to a device connected to a computer network and that is used to identify the device in various forms of electronic communication such as the World Wide Web, e-mail or Usenet. Hostnames may be simple names consisting of a single word of phrase, or they may include the name of a Domain Name System (DNS) domain at the end, that is separated from the host specific label by a full stop (dot). In the latter form, a hostname is also called a domain name. If the domain name is completely specified including a top-level domain of the Internet, the hostname is said to be a fully qualified domain name (FQDN).

Hostnames that include DNS domains are often stored in the Domain Name System together with IP addresses of the host they represent for the purpose of mapping the hostname to an address, or the reverse process.