By L. E. Sigler (auth.)

ISBN-10: 3540901957

ISBN-13: 9783540901952

ISBN-10: 3662267381

ISBN-13: 9783662267387

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**Additional info for Algebra**

**Example text**

An expression such as 2 + 5X + 9X 2 or t + tX + 1/X 5 is known as a polynomial. While we shall study polynomial rings in detail in Chapter 5 many aspects of polynomials are familiar and intuitive enough we can use them now as examples. • , a. E Q, nE N}, is a ring with the usual school algebra way of adding and multiplying polynomials. Find (t], [X], [Q u {X}]. 49 2. 5 Morphisms In this section the structure-preserving functions, morphisms, are defined and investigated. J') be two rings andf:R--+ R' a function.

S, t)(u The right distributive theorem is verified similarly. This completes the verification that (71. , +, ·, (0, 0), (1, 1)) is a unitary commutative ring. The construction in the previous example motivates the following definition of a product ring. Definition. Given rings (R', + ', ·', e') and (R", + ", ·", e") we define the product of the two rings to be ( R' x R", +, ·, (e', e")) in which + and ·are defined by (x', x") + (y', y") = (x' +' y', x" +" y") (x', x") · (y', y") = (x' ·' y', x" ·" y").

When we know the one value, v(O) = 67, we know the entire function, the entire nullary operation. We have seen in the previous example, following the consequences ofthe definition, that knowing a nullary operation v: S0 -+ S is simply knowing one value, v(O). A nullary operation, in effect, picks from S one element, the value of v(O). Conversely, any choice of a single element from a set, or any designation of a single element from a set can be expressed in the form of a nullary operation on the set.

### Algebra by L. E. Sigler (auth.)

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