Page 164 - DMTH403_ABSTRACT_ALGEBRA

Unit 15: Subrings

(ii) 0 E S, and Notes
(iii) for each a E S, - a E S.

Even this definition can be improved upon. For this, recall from Unit 3 that (S, f )  (R, +)

ifa – b E S whenever a, b E S. This observation allows us to give a set of conditions for a
subset to be a subring, which are easy to verify.

15.3 Keyword

Subring: Let (R, +, .) be a ring and S be a subset of R. We say that S is a subring of R, if (S, +, .) is
itself a ring, i.e., S is a ring with respect to the operations on R.

15.4 Review Questions

   a 0       a 0   
1. Show that S =   a,b Z .  is a subring of R =   a,b R .  Does S have a unit


   0 b       0 b   
element?
If yes, then is the unit element the same as that of R?
2. For any ring R, show that {0} and R are its subrings.
3. Show that if A is subring of B and B is a subring of C, then A is a subring of C.
4. Give an example of a subset of Z which is not a subring.



a,
5. Show that   3 and 0,2,4 are proper ideal of Z .
6
Answers: Self Assessment
1. (b) 2. (b) 3. (a)

15.5 Further Readings

Books Dan Saracino: Abstract Algebra; A First Course.
Mitchell and Mitchell: An Introduction to Abstract Algebra.
John B. Fraleigh: An Introduction to Abstract Algebra (Relevant Portion).

Online links www.jmilne.org/math/CourseNotes/
www.math.niu.edu
www.maths.tcd.ie/

archives.math.utk.edu

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