Investiga tion 9

Subrings, Extensions, and Direct Sums

Focus Questions

By the end of this investigation, you should be able to give precise and thorough

answers to the questions listed below. You may want to keep these questions in mind

to focus your tho ughts as you complete the investigation.

• What is a subring? What conditions must be veriﬁed in order to show that a subset

of a ring is a subring?

• In what ways are a ring and all of its subrings guaranteed to be similar? In what

ways can a ring and its subrings be d ifferent?

• What is a ﬁeld extensio n, and how can ﬁeld extension s be used to construct larger

rings from smaller ones?

• What is a d irect sum, and h ow can direct sums be used to construct larger rings

from smaller ones?

• How are the properties of ﬁeld extensions and direct sums related to the properties

of the individual rings used to construct them?

Preview Activity 9.1. Throughout mathema tics, the relationship between mathematical objects and

their sub-obje cts is of central importance. For instanc e, in linear algebra, we study vector spaces

and their subspaces. In discr ete mathematics, many graph theory problems can be solved by ﬁnding

a subgraph that is optimal in some sense. Furthermore, many other a pplied optimizatio n problems

involve minimizing or maximizing a certain f unction subject to certain constraints. The se constraints

deﬁne what is known as a feasible region, which is nothing more than a subset of the space of all

possible so lutions.

In light of these examples and our recent investigations of rings, it seems natural that we would

be interested in deﬁning and characterizing subrings. To begin thinking along these lines, we will

consider a set of numbers that is larger than Q but smaller than R. The set, denoted Q(

√

2), is

deﬁned as follows:

Q(

√

2) = {a + b

√

2 : a, b ∈ Q}

(a) Show that Q ⊂ Q(

√

2) ⊂ R (that is, Q is a proper subset of Q(

√

2), which is a proper subset

of R).

(b) With addition and multiplication d eﬁned as in R, which of the ring axioms does Q(

√

2)

satisfy? Give a brief explanation to justify your answer for each axiom.

105

Get *Abstract Algebra* now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.