# \ #
# "We are given:" #
# "a)" \ R \ "is a commutative ring with identity." #
# "b)" \ S \ "is an integral domain." #
# "c)" \ f: R rarr S \ "is a non-trivial ring homomorphism." #
# "We are asked to show:" #
# "d)" \ ker(f)\ "is a prime ideal of" \ R. #
# \ #
# "Proof." #
# "1) Because" \ f \ "is at least a ring homomorphism, we have:" #
# \qquad \qquad \qquad \qquad \qquad \qquad R/{ ker(f) } \ ~~ \ Im(f) \ sube \ S. #
# "2) Because" \ f \ "is at least a ring homomorphism, we have:" #
# \qquad \qquad \qquad \qquad Im(f) \ "is a ring, and so a sub-ring of" \ S. #
# "3) Because" \ S \ "is an integral domain, we have, by definition:" #
# \qquad \qquad \qquad \ \ S \ "has no elements that are zero-divisors". #
# "4) Because" \ Im(f) \ "is at least a subset of" \ S, "we have, a fortiori:" #
# \qquad \qquad \quad Im(f) \ "has no elements that are zero-divisors". #
# "5) Because" \ Im(f) \ "is a sub-ring of" \ S, "by (2); and has no" #
# "elements that are zero-divisors, by (4); we have, by definition:" #
# \qquad \qquad \qquad \qquad \qquad \qquad Im(f) \ \"is an integral domain". #
# "6) Now we are almost finished. From (1) and (5), we have:" #
# \qquad R/{ ker(f) } \ ~~ \ Im(f) \qquad "and" \qquad Im(f) \ \ "is an integral domain". #
# \qquad \qquad \qquad \qquad :. \qquad \qquad R/{ ker(f) } \ ~~ \ "an integral domain". #
# "As isomorphisms respect integral domains:" #
# \qquad \qquad \qquad \qquad :. \qquad \qquad R/{ ker(f) } \ \ "is an integral domain." \qquad \qquad \qquad \qquad ("A") #
# "7) Recall that, for any ring" \ R, \ "and any ideal" \ I \ \ "of" \ R: #
# \qquad R/I \ \ "is an integral domain" \ hArr \ I \ \ "is a prime ideal of" \ R. \qquad ("B") #
# "8) So, as" \ \ ker(f) \ \ "is an ideal for any ring homomorphism" \ f, \ "with (A) and (B), we have:" #
# \qquad \qquad \qquad \qquad \qquad ker(f) \ \ "is a prime ideal of" \ R. \qquad \qquad square #
# \ #
# "Additional Remark (Pedagogic)" #
# "The statement in (B) above, is a fundamental fact about rings:" #
# \qquad R/I \ \ "is an integral domain" \ hArr \ I \ \ "is a prime ideal of" \ R. \qquad ("B") #
# "It has a paired fact, and I like to state the two together:" #
# \qquad R/I \ \ "is an integral domain" \ hArr \ I \ \ "is a prime ideal of" \ R. \qquad \quad ("B") #
# \qquad R/I \ \ "is a field" \qquad \qquad \qquad \qquad \qquad \ \ \ hArr \ I \ \ "is a maximal ideal of" \ R. \ ("C") #
# "These are fundamental facts about rings, and are used," #
# "frequently, and powerfully, everywhere !!" #
