{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

College Algebra Exam Review 67

# College Algebra Exam Review 67 - 77 1.11 RINGS AND FIELDS...

This preview shows page 1. Sign up to view the full content.

1.11. RINGS AND FIELDS 77 Example 1.11.5. (a) Z is a subring of Q . (b) The ring of 3-by-3 matrices with rational entries is a subring of the ring of 3-by-3 matrices with real entries. (c) The set of upper triangular 3-by-3 matrices with real entries is a subring of the ring of all 3-by-3 matrices with real entries. (d) The set of continuous functions from R to R is a subring of the ring of all functions from R to R . (e) The set of rational–valued functions on a set X is a subring of the ring of real–valued functions on X . A second way in which two rings may be related to one another is by a homomorphism . A map f W R ! S between two rings is said to be a homomorphism if it “respects” the ring structures. More explicitly, f must take sums to sums and products to products. In notation, f .a C b/ D f .a/ C f .b/ and f .ab/ D f .a/f .b/ for all a; b 2 R . A bijective homomorphism is called an isomorphism . Example 1.11.6. (a) The map f W Z ! Z n defined by f .a/ D OEaŁ is a homomor- phism of rings, because f .a C b/ D OEa C D OEaŁ C
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Ask a homework question - tutors are online