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

b) Show that the abelian group (Z[ - five ] , +) is a ring under
multiplication (a
+ b - five )(c + d - five ) = ac – 5bd + (ad + bc) - five .
7) a) describe i) a division ring ii) an unit element iii) an integral domain
b) Show that the abelian group (Z[i] , +) is a ring under
multiplication
(a + bi)(c + di) = ac – bd + (ad + bc) i, for all a + bi ,c + di ? Z[i].
8a) Let R be a ring with identity one and (ab)2 = a2b2, for all a, b ? R. Show
that R is commutative.
b)Show that the abelian group (Zn , +n) is a commutative ring with
identity one under multiplication modulo n operation.
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9 a) Show that a ring R is commutative if and only if (a + b)2 = a2 + 2ab +
b2, for all a, b ? R.
b) Show that Z[i] = {a + ib ?a , b ? Z}, the ring of Gaussian integers,
is an integral domain.
10 a) Show that a commutative ring R is an integral domain if and only if
a , b , c ? R, a ? 0, ab = ac ? b = c.
b) Prepare addition modulo four and multiplication modulo four tables. obtain
all invertible elements in Z4.
11 a) Show that a commutative ring R is an integral domain if and only if
a , b ? R, ab = 0 ? either a = 0 or b = 0.
b) Prepare addition modulo five and multiplication modulo five tables. obtain
all invertible elements in Z5.
12 a) Let R be a commutative ring. Show that the cancellation legal regulations with
respect to multiplication holds in R if and only if a , b ? R, ab = 0
? either a = 0 or b = 0.
b) Prepare a multiplication modulo six table for a ring (Z6 , +6 , ×6).
Hence obtain all zero divisors and invertible elements in Z6.
13 a) For n > 1, show that Zn is an integral if and only if n is prime.
b) Let R =
? ? ?
? ? ?
? ??
?
??
?
-
: z, w C
w z
z w be a ring under addition and
multiplication, where C = {a + ib ?a , b ? R}. Show that R is a
divison ring.
14 a) Prove that every field is an integral domain. Is the converse true?
Justify.
b) Which of the subsequent rings are fields? Why?
i) (Z , + , ×) ii) (Z5 , +5 , ×5) iii) (Z25 , +25 , ×25).
15) a) Prove that every finite integral domain is a field.
b) Which of the subsequent rings are integral domains? Why?
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i) (2Z , + , ×) ii) (Z50 , +50 , ×50) iii) (Z17 , +17 , ×17).
16 a) Prove that a Boolean ring is a commutative ring.
b) provide an example of a division ring which is not a field.
17 a) for n > 1, show that Zn is a field if and onle if n is prime.
b) Let R = {a + bi + cj + dk ?a, b, c, d ? R}, where i2 = j2 = k2 = –1
, ij = k = –ji , jk = i = –kj , ki = j = –ik. Show that every nonzero
element of R is invertible.
18 a) If R is a ring and a, b ? R then prove or disprove (a + b)2 = a2 +
2ab + b2.
b) Show that R+ , the set of all positive reals forms a ring under the
subsequent binary operations :
a ? b = ab and a ?? b =
log5b
a , for all a,b ? R+.
19 a) describe i) a ring ii) a Boolean ring iii) an invertible
element.
b) Let p be a prime and (pZ , +) be an abelian group under usual
addition, show that (pZ, + , ??) is a commutative ring with
identity element p where a ?? b =
p
ab , for all a , b ? pZ.
20) a) describe i) a ring with identity element ii) a commutative ring
iii) a zero divisor.
b) Show that R+ , the set of all positive reals forms a ring under the
subsequent binary operations :
a ? b = ab and a ?? b =
log7b
a , for all a,b ? R+

Earning:   Approval pending.