# \ #
# "Proofs." #
# "i) Because 5 is a prime of" \ ZZ ":" \ \ ZZ_5 \ "is a field [basic fact of number" #
# "theory, or algebra]."#
# "As fields are particular examples of integral domains," \ ZZ_5 \ "will, a fortiori, contain no zero-divisors."#
# "Also, this can actually -- and easily -- be checked by direct" #
# "computation in" \ ZZ_5 \ "(and which also contains only 5 elements," #
# "of course.)" #
# "Thus the request in (i) is impossible to satisfy."#
# "ii) I take it that by" \ \ C[0, 1], "is meant the set of continuous," #
# "real functions on" \ [0, 1]. "Taking that to be the case, let:"#
# \qquad \qquad \qquad f(x) \ = \ 1; \ "a constant function on" \ [0, 1]. #
# "Clearly:" \qquad \qquad \qquad \qquad \qquad f(x) \ in C[0, 1]. #
# "Now let:" \qquad \qquad \qquad \qquad \qquad g(x) \ in C[0, 1]. #
# "Suppose:" \qquad \qquad \quad f(x) \cdot g(x) = 0 \qquad "[constant function" \ 0"]". #
# "Thus:" \qquad \qquad \qquad \qquad \ \ \ 1 \cdot g(x) \ = \ 0 \qquad "[constant function" \ 0"]". #
# "So:" \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ g(x) = 0 \qquad "[constant function" \ 0"]". #
# "Thus:" \quad \ \ f(x) \cdot g(x) = 0 \ \ rArr \ \ g(x) = 0 \quad "[constant function" \ 0"]". #
# "As" \ g(x) \ "is taken as an arbitrary element of" \ C[0, 1], \ "we have that" \ f(x) \ "is not a zero-divisor of" \ C[0, 1]." #
# "[Exercise (Optional, Of course !!):"#
# \qquad \qquad "What are all the zero-divisors, if any, of" \ C[0, 1] ? "]" #
# "iii) Recall, by definition:" #
# "A ring" \ R \ "is an integral domain" hArr R \ "contains no zero-divisors."#
# "So, a fortiori, any sub-ring, or even any subset, of an integral" #
# "domain contains no zero-divisors, either." #
# "Thus, any sub-ring of an integral domain is an integral domain." #
# "Thus the request in (iii) is impossible to satisfy."#
