Maximal Ideal Definition, Examples, and Relationship to Fields in Abstract Algebra

What is the definition of a maximal ideal in a commutative ring? A maximal ideal A of a commutative ring R is a proper ideal of R with the property that if B is an ideal of R containing A, then either B = A or B = R. For examples of maximal ideals, we create the subring lattice for the ring ℤ12 to find maximal ideals of ℤ12. Here's a VERY important Theorem: In a commutative ring R with unity, the factor ring R/A is a field if and only if A is a maximal ideal. We relate this to a previous example. The factor ring ℤ3[x]/＜x^2+1＞ is a field with 9 elements. This means A=＜x^2+1＞ is a maximal ideal in ℤ3[x]. This is related to the fact that f(x)=x^2+1 has no zeros in ℤ3, so it is irreducible over ℤ3. Links and resources =============================== 🔴 Subscribe to Bill Kinney Math: https://www.youtube.com/user/billkinn... 🔴 Subscribe to my Math Blog, Infinity is Really Big: https://infinityisreallybig.com/ 🔴 Follow me on Twitter:   / billkinneymath   🔴 Follow me on Instagram:   / billkinneymath   🔴 You can support me by buying "Infinite Powers, How Calculus Reveals the Secrets of the Universe", by Steven Strogatz, or anything else you want to buy, starting from this link: https://amzn.to/3eXEmuA. 🔴 Check out my artist son Tyler Kinney's website: https://www.tylertkinney.co/ 🔴 Desiring God website: https://www.desiringgod.org/ AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.