“ring”的概念、定义、翻译、参考文献-科学参考

ring

Mathematics

Sets with two binary operations, often called addition and multiplication, occur commonly in mathematics and sometimes share many of the same properties. One such set of properties is specified in the definition of a ring: a ring is a set R, closed under two operations called addition and multiplication, such that
1. (i) for all a, b, and c in R, a+(b + c) = (a + b) + c,
2. (ii) for all a and b in R, a + b = b + a,
3. (iii) there is an element 0 in R such that a+0 = a for all a in R,
4. (iv) for each a in R, there is an element −a in R such that a+(−a) = 0,
5. (v) for all a, b, and c in R, a(bc) = (ab)c,
6. (vi) for all a, b, and c in R, a(b + c) = ab + ac and (a + b)c = ac + bc.
The element 0 guaranteed by (iii) is an additive identity. It can be shown to be unique and has the extra property that a0 = 0 for all a in R; it is called zero. Also, for each a, the element −a guaranteed by (iv) is unique and is the negative or additive inverse of a. The ring is a commutative ring if it is further true that
1. (vii) for all a and b in R, ab = ba,
  
  and it is a commutative ring with identity if also
2. (viii) there is an element 1 (≠0) such that a1 = a for all a in R.
The element 1 guaranteed by (viii) is a multiplicative identity. It can be shown to be unique and is referred to as ‘one’.

Further properties may be required for other types of ring such as integral domains and fields. Examples of rings include the set of 2 × 2 real matrices and the set of all even integers, each with the appropriate addition and multiplication. Another example is ℤ_n, the set of integers with addition and multiplication modulo n.

A ring may be denoted by (R,+,×) when it is necessary to be clear about the ring’s operations. But it is sufficient to refer simply to the ring R when the operations intended are clear.

Chemistry

A closed chain of atoms in a molecule. In compounds, such as naphthalene, in which two rings share a common side, the rings are fused rings. Ring closures are chemical reactions in which one part of a chain reacts with another to form a ring, as in the formation of lactams and lactones.

Computer

1. An algebraic structure R on which there are defined two dyadic operations, normally denoted by + (addition) and • or juxtaposition (multiplication). With respect to addition, R is an abelian group,
$〈 R, + 〉$
i.e. + is commutative and associative. With respect to multiplication, R is a semigroup,
$〈 R, • 〉$
i.e. • is associative. Further, multiplication is distributive over addition.
Certain kinds of rings are of particular interest:
1. (a) if multiplication is commutative the ring is called a commutative ring;
2. (b) if 〈R, •〉 is a monoid, the ring is called a ring with an identity;
3. (c) a commutative ring with an identity, and having no nonzero elements x and y with the property that x • y = 0, is said to be an integral domain;
4. (d) a commutative ring with more than one element, and in which every nonzero element has an inverse with respect to multiplication, is called a field.
The different identity elements and inverses, when these exist, can be distinguished by talking in terms of additive identities (or zeros), multiplicative identities (or ones), additive inverses, and multiplicative inverses.
The concept of a ring provides an algebraic structure into which can be fitted such diverse items as the integers, polynomials with integer coefficients, and matrices; on all these items it is customary to define two dyadic operations.
2. Another name for circular list, but more generally applied to any list structure where all sublists as well as the list itself are circularly linked.
3. In network topology, a ring network is a closed-loop network that does not require terminators. A token ring topology is physically cabled as a star, with a logical ring maintained at the hub. When a workstation connects to the hub, the ring is extended out to the workstation and back to the hub.

单词	ring
释义	ring Mathematics Sets with two binary operations, often called addition and multiplication, occur commonly in mathematics and sometimes share many of the same properties. One such set of properties is specified in the definition of a ring: a ring is a set R, closed under two operations called addition and multiplication, such that (i) for all a, b, and c in R, a+(b + c) = (a + b) + c, (ii) for all a and b in R, a + b = b + a, (iii) there is an element 0 in R such that a+0 = a for all a in R, (iv) for each a in R, there is an element −a in R such that a+(−a) = 0, (v) for all a, b, and c in R, a(bc) = (ab)c, (vi) for all a, b, and c in R, a(b + c) = ab + ac and (a + b)c = ac + bc. The element 0 guaranteed by (iii) is an additive identity. It can be shown to be unique and has the extra property that a0 = 0 for all a in R; it is called zero. Also, for each a, the element −a guaranteed by (iv) is unique and is the negative or additive inverse of a. The ring is a commutative ring if it is further true that (vii) for all a and b in R, ab = ba, and it is a commutative ring with identity if also (viii) there is an element 1 (≠0) such that a1 = a for all a in R. The element 1 guaranteed by (viii) is a multiplicative identity. It can be shown to be unique and is referred to as ‘one’. Further properties may be required for other types of ring such as integral domains and fields. Examples of rings include the set of 2 × 2 real matrices and the set of all even integers, each with the appropriate addition and multiplication. Another example is ℤ_n, the set of integers with addition and multiplication modulo n. A ring may be denoted by (R,+,×) when it is necessary to be clear about the ring’s operations. But it is sufficient to refer simply to the ring R when the operations intended are clear. Chemistry A closed chain of atoms in a molecule. In compounds, such as naphthalene, in which two rings share a common side, the rings are fused rings. Ring closures are chemical reactions in which one part of a chain reacts with another to form a ring, as in the formation of lactams and lactones. Computer 1. An algebraic structure R on which there are defined two dyadic operations, normally denoted by + (addition) and • or juxtaposition (multiplication). With respect to addition, R is an abelian group, $〈 R, + 〉$ i.e. + is commutative and associative. With respect to multiplication, R is a semigroup, $〈 R, • 〉$ i.e. • is associative. Further, multiplication is distributive over addition. Certain kinds of rings are of particular interest: (a) if multiplication is commutative the ring is called a commutative ring; (b) if 〈R, •〉 is a monoid, the ring is called a ring with an identity; (c) a commutative ring with an identity, and having no nonzero elements x and y with the property that x • y = 0, is said to be an integral domain; (d) a commutative ring with more than one element, and in which every nonzero element has an inverse with respect to multiplication, is called a field. The different identity elements and inverses, when these exist, can be distinguished by talking in terms of additive identities (or zeros), multiplicative identities (or ones), additive inverses, and multiplicative inverses. The concept of a ring provides an algebraic structure into which can be fitted such diverse items as the integers, polynomials with integer coefficients, and matrices; on all these items it is customary to define two dyadic operations. 2. Another name for circular list, but more generally applied to any list structure where all sublists as well as the list itself are circularly linked. 3. In network topology, a ring network is a closed-loop network that does not require terminators. A token ring topology is physically cabled as a star, with a logical ring maintained at the hub. When a workstation connects to the hub, the ring is extended out to the workstation and back to the hub.
随便看	biotic factor biotic index biotic potential biotin biotite Biot, Jean Baptiste Biot, Jean-Baptiste (1774–1862) Biot number biotope bioturbation biotype Biot–Savart law biounlimiting elements biozone bipartite graph bipedalism biphase PCM biplot bipolar cell bipolar distribution bipolar group bipolar integrated circuit bipolar junction transistor bipolar logic circuit bipolar nebula