Ring Theory and Linear Algebra-1 in BSC(H) Mathematics Semester-IV
Explore the concepts of ideals and factor rings in ring theory alongside normal subgroups in group theory. Delve into the definitions, theorems, and applications, providing a solid foundation in abstract algebra.
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BSC(H)MATHEMATICS SEMESTER-IV SUBJECT: RING THEORY AND LINEAR ALGEBRA-1 DEPARTMENT: MATHEMATICS
CHAPTER-14 CREDIT: J. GALLIAN, 4THEDITION CONTEMPORARY ABSTRUCT ALGEBRA
Normal subgroups play a special role in group theorythey permit us to construct factor groups. In this chapter, we introduce the analogous concepts for rings ideals and factor rings.
Definition Ideal A subring A of a ring R is called a (two-sided) ideal of R if for every r [ R and every a [ A both ra and ar are in A. An ideal A of R is called a proper ideal of R if A is a proper subset of R.
Theorem 14.1 (Ideal Test) A nonempty subset A of a ring R is an ideal of R if 1. a - b A whenever a, b A. 2. ra and ar are in A whenever a A and r R.
Theorem 14.2 Existence of Factor Rings Let R be a ring and let A be a subring of R. The set of cosets {r + A | r R} is a ring under the operations (s + A) + (t + A) = s + t + A and (s + A)(t + A) = st + A if and only if A is an ideal of R.
let ? Where ? + ? + ? + ? = ? + ? + ? ? + ? ? + ? = ?? + ? To show : ? is an ideal. Given : ? is a sub ring of ?. For showing ? is ideal it is enough to show ??,?? ? where every ? ? ,? ?. ?= ? + ?: ? ?
Let ? ? , ? ? ? + ? = ? = ???? ??? ?& ? + ? ? ? ?? + ? = ? + ? (? + ?) = ???? ??? ? = ???? ??? ?= ? ?? + ? = ? ?? ? Similarly ?? ? ? is ideal of ? ? + ?
Converse : Let ? be the ideal of ?. To show : ? ?= ? + ? ? ? is a ring. Since ? is ring , ? ? ? Let ?1+ ? ,?2+ ? , ?3+ ? ? (i) ?1+ ? + ?2+ ? = ?1+ ?2 + ? ? (ii) ?1+ ? + ?2+ ? = ?1+ ?2 + ? ? ?( ?1+ ?2 ?) = ?2+ ?1 + A ( ?,+ ?? ????.) = ?2+ ? + ?1+ ?
(iii) ?1+ ? + ?2+ ? + ?3+ ? = ?1+ ?2 + ? + ?3+ ? ?1+ ?2 + ?3 + ? + ? ( ?,+ ?? ?????.) ?1+ ? + ( ?2+ ?3 + ?) ?1+ ? + ( ?2+ ? + ?3+ ? ) (iv) ?1+ ? + 0 + ? = ?1+ 0 + ? = ?1+ ? ||?? 0 + ? + ?1+ ? = ?1+ ? ( 0 is zero of ?) 0 + ? = ? is zero of ? ? = = ?1+ ?2+ ?3 = =
(?) since ?1 ? and ? is ring ?1 ? And ?1+ ? + ( ?1) + ? = 0 + ? = ? ( ?1) + ? is negative of ? ?1+ ? + ?2+ ? = ?1+ ?2 + ? ? ? ? (viii) Associative ( do yourself)
(viii) ?1+ ? = ?1?2+ ?3 + ? = ?1+ ?2+ ?1?3 + ?( ? ?? ???? ) = ?1+ ?2+ ? + ?1?3+ ? = ?1+ ? ( ?2+ ? + ?1+ ? + ( ?3+ ? (ix) Similarly ?2+ ? + ?3+ ? = ?1+ ? ?2+ ?3 + ? ?1+ ? ?3+ ? + ( ?1+ ? + ( ?2+ ? ) ? ?3+ ? ?3+ ? = ?2+ Hence R/A is a ring.
