North Maharashtra University 2008 B.Sc Mathematics S.YBSc MTH – 212 (B) (Computational Algebra) - Question Paper

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NORTH MAHARASHTRA UNIVERSITY, JALGAON

(New Syllabus w.e.f. June 2008)
Class : S.Y. B. Sc. Subject : Mathematics
Paper : MTH – 212 (B) (Computational Algebra)
(Calculus of Several Variables)

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1 : ques. of two marks
1) In a ring (Z , ? , ??), where a ? b = a + b – one and a ?? b = a + b – ab , for
all a , b ? Z, obtain zero element and identity element.
2) describe an unit. obtain all units in (Z6 , +6 , ×6).
3) describe a zero divisor. obtain all zero divisors in (Z8 , +8 , ×8).
4) Let R be a ring with identity one and a ? R. Show that
i) (–1)a = –a ii)(–1) (–1) = 1
5) Let R be a commutative ring and a , b ? R. Show that (a – b)2 = a2 – 2ab
+ b2.
6) Let (Z[ - five ] , + , ??) be a ring under usual addition and multiplication of
elements of Z[ - five ]. Show that Z[ - five ] is a commutative ring . Is two +
3 - five a unit in Z[ - five ]?
7) Let m ? (Zn , +n , ×n) be a zero divisor. Show that m is not relatively
prime to n, where n > 1.
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8) If m ? (Zn , +n , ×n) is invertible then show that m and n are relatively
prime to n, where n > 1.
9) Let n > one and 0 < m < n. If m is relatively prime to n then show that
m ? (Zn , +n , ×n) is invertible.
10) Let n > one and 0 < m < n. If m is not relatively prime to n then show that
m ? (Zn , +n , ×n) is a zero divisor.
11) Show that a field has no zero divisors.
12) Let R be a ring in which a2 = a, for all a ? R. Show that a + a = 0, for all a
? R.
13) Let R be a ring in which a2 = a, for all a ? R. If a , b ? R and a + b = 0,
then show that a = b.
14) Let R be a commutative ring with identity 1. If a , b are units in R then
show that a-1 and ab are units in R.
15) In (Z12 , +12 , ×12) obtain (i) (3 )2 +12 (5 )-2 (ii) ( seven )-1 +12 eight .
16) In (Z12 , +12 , ×12) obtain (i) (5 )-1 – seven (ii) (11)-2 +12 five .
2 : Multiple option ques. of one marks
select the accurate choice from the provided choices.
1) R = { ± 1, ± 2, ± 3, - - - } is not a ring under usual addition and
multiplication of integers because - - -
a) R is not closed under multiplication
b) R is not closed under addition
c) R does not satisfy associativity w.r.t. addition
d) R does not satisfy associativity w.r.tmultiplication
2) Number of zero divisors in (Z6 , +6 , ×6) = - - -
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a) 0 b) one c) two d) 3
3) (Z43 , +43 , ×43) is - - -

Earning:   Approval pending.