Lemma 214 an ideal p of a unital commutative ring r

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Lemma 2.14 An ideal P of a unital commutative ring R is a prime ideal of R if and only if the quotient ring R/P is an integral domain. Proof Let P be an ideal of R . Then P is prime if and only if P is a proper ideal of R and, given elements x and y that do not belong to the ideal P , the product xy of those elements does not belong to P . Now an element x of R belongs to an ideal P if and only if the image x + P of x under the quotient homomorphism is the zero element of R/P . It follows that the ideal P of R is prime if and only if R/P is a commutative ring with a non- zero multiplicative identity element P + 1 R in which the product of any two non-zero elements of the quotient ring R/P is always a non-zero element of that quotient ring. It then follows from the definition of integral domains that an ideal P of a unital commutative ring R is prime if and only if R/P is an integral domain. Lemma 2.15 Every maximal ideal of a unital commutative ring R is a prime ideal of R . Proof Every field is an integral domain. The result therefore follows imme- diately from Lemma 2.13 and Lemma 2.14. Lemma 2.16 The zero ideal { 0 R } of a unital commutative ring R is a prime ideal of R if and only if R is an integral domain. Proof The zero ideal of a unital commutative ring R is a proper ideal of R . It is therefore prime if and only if the product of non-zero elements of R is always non-zero, and thus is prime if and only if R is an integral domain. Lemma 2.17 An integral domain with only finitely many elements is a field. Proof Let R be an integral domain with only finitely many elements, and let x be a non-zero element of R . Then x determines an injective function λ x : R R from R to itself, where λ x ( y ) = xy for all y R . This injective function must be surjective, because R is a finite set. Therefore there exists some element y of R such that λ x ( y ) = 1 R , where 1 R is the multiplicative identity element of R . Then xy = 1 R . This proves that every non-zero element of the integral domain R is a unit of R , and therefore R is a field. 21
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Lemma 2.18 Let R be a unital commutative ring, and let P be a prime ideal of R . Suppose that the number of cosets of P in R is finite. Then P is a maximal ideal of R . Proof The quotient ring R/P is an integral domain, because the ideal P is prime (Lemma 2.14). This integral domain has only finitely many elements because those elements are the cosets of P in R . Therefore R/P is a field (Lemma 2.17), and thus the prime ideal P is a maximal ideal (Lemma 2.13). Lemma 2.19 Let x be an element of an integral domain R . Then x is a prime element of R if and only if the principal ideal ( x ) generated by x is a non-zero prime ideal of R . Proof Let x be a prime element of R . Then x is non-zero and is not a unit of R . It follows that ( x ) is a non-zero proper ideal of R . Let y and z be elements of R satisfying yz ( x ). Then x | yz . Therefore either x | y or x | z , because x is a prime element of R , and thus either y ( x ) or z ( x ). Thus the principal ideal ( x ) is a non-zero prime ideal of R .
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