A subset S of a ring R is said to be multiplicative if ab∈S∀a,b∈S. Show that P is a prime ideal of a commutative ring R if and only if R−P is multiplicative.
Adv.
Let I be the principal ideal ⟨x⟩ of the ring Z[x]. Prove that I is prime but not maximal.
Let R be a commutative ring with identity in which the set of all non-units forms an ideal M. Show that M is a maximal ideal of R and it contains every proper ideal of R. Also show that for each r∈R either r or 1−r is a unit.
Let R be a commutative ring. Show that a|b⇔Ra⊃Rb,∀a,b∈R. Further if R contains the identity, show that for any a∈R, if Ra is maximal, then a is an irreducible element of R.
Let R be a non-trivial ring with identity in which every subring is an ideal. Show that R≅Z or Zn for some n.
Write parametric equation of hypocycloid, epicycloid, epitrochoid, trochoid.
Let C be circle |z|=R(R>1) oriented counterclockwise. Show that
|∫CLogz2z2dz|≤4π(π+logRR) and hence lim .
Without evaluating the integral, show that \left | \int_C \frac {1}{\overline{z}^2 + \overline{z} +1}dz \right| \le \frac{9\pi}{16} where C is the arc of circle |z|=3 from z=3 to z=3i lying in the first quadrant.
Find the zeros and poles of
f(z) = \frac{1}{(z-a)(z-b)(e^{z-a} -1)}
and hence
\int_{-\infty}^{\infty} \frac{1}{(x-a)(x-b)(e^{x-a} -1)}dx
where a,b are real .
Let f be the function defined by f(z)=\frac{(z+5)(2z+1)}{z^2-4} . Find the zero and poles of f,\frac{1}{f}, f^\prime and \frac{f^\prime}{f} .
Show that the function f defined by the series f(z)=\sum ^{\infty}_{n=1}\frac{1}{(z+n)^2} is meromorphic on every bounded subset of \mathbb{C}, and find the residues at its poles.
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