Prime and Maximal Ideals | Maths | Chegg Tutors

An ideal ℑ in a commutative ring ℛ is said to be a prime ideal if, for arbitrary element r, s of ℛ, the fact that r · s ∈ ℑ implies either r ∈ ℑ or s ∈ ℑ.
Example 15. In the ring ℤ,
(a) The ideal J = {7r : r ∈ ℤ}, also written as J = (7), is a prime ideal since if a · b ∈ J either 7|a or 7|b; hence, either a ∈ J or b ∈ J.
(b) The ideal K = {14r : r ∈ ℤ} or K = (14) is not a prime ideal since, for example, 28 = 4 · 7 ∈ K, but neither 4 nor 7 is in K.
Example 15 illustrates
Theorem VII. In the ring ℤ a proper ideal ℑ = {mr : r ∈ ℤ, m ≠ 0} is a prime ideal if and only if m is a prime integer.
A proper ideal ℑ in a commutative ring ℛ is called maximal if there exists no proper ideal in ℛ which properly contains ℑ.
Example 16.
(a) The ideal J of Example 15 is a maximal ideal in ℤ since the only ideal in ℤ which properly contains J is ℤ itself.
(b) The ideal K of Example 15 is not a maximal ideal in ℤ since K is properly contained in J, which, in turn, is properly contained in ℤ.

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About Chris W., Maths tutor on Chegg Tutors:

University of Pennsylvania, Class of 2007
Math, Computer Science major

Subjects tutored: Linear Algebra, Pre-Algebra, Applied Mathematics, Calculus, Number Theory, Algebra, Statistics, SSAT (math), Linear Programming, Computer Certification and Training, SAT II Mathematics Level 2, Information Technology, Geometry (College Advanced), SAT (math), PSAT (math), Set Theory, Discrete Math, Basic Math, Trigonometry, ACT (math), LaTeX, SAT II Mathematics Level 1, Numerical Analysis, Web Design, Pre-Calculus, Computer Science, Geometry, Software Engineering, and GRE

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Over 7 years of experience teaching math at 3 universities and a community college. Courses ranged from Intermediate Algebra to Calculus II and class sizes varied from 2 to over 200 students. Tutoring since 2000 formally and informally, individually and in groups, for courses from Geometry to Differential Equations. Please note that I generally will not be available for audio and video in live lessons but my experience has been that audio and video aren't really needed.

EXTRACURRICULAR INTERESTS
Hiking, reading, video games, playing the guitar and piano.

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[consecrator]

Please discuss about how to make or define a map, ?

[melaniekarlsen]

You repeatedly said "I is a principle ideal but not a maximal ideal" towards the end, but I think that you meant to say "I is a prime ideal but not a maximal ideal", didn't you?
Thanks for the example.