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8: An Introduction to Rings
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/08%3A_An_Introduction_to_Rings
7.4: Modular Arithmetic
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/07%3A_New_Page/7.04%3A_New_Page
We define the sum and product of congruence classes in $\mathbb{Z}/n\mathbb{Z}$ via $[a]_n + [b]_n:= [a+b]_n \quad \text{and} \quad [a]_n \cdot [b]_n:= [a\cdot b]_n.$ However, it is possible for \...We define the sum and product of congruence classes in $\mathbb{Z}/n\mathbb{Z}$ via $[a]_n + [b]_n:= [a+b]_n \quad \text{and} \quad [a]_n \cdot [b]_n:= [a\cdot b]_n.$ However, it is possible for $[a]_n\cdot[b]_n = [0]_n$ even when $[a]_n \neq [0]_n$ and $[b]_n \neq [0]_n$ . If $n\in \mathbb{N}$ such that $n$ is not prime, then there exists $[a]_n, [b]_n \in \mathbb{Z}/n\mathbb{Z}$ such that $[a]_n\cdot[b]_n = [0]_n$ while $[a]_n \neq [0]_n$ and $[b]_n \neq [0]_n$ .
4.1: Cyclic Groups
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/04%3A_Families_of_Groups/4.01%3A_Cyclic_Groups
Recall that if $G$ is a group and $g\in G$ , then the cyclic subgroup generated by $g$ is given by $\langle g\rangle =\{g^k\mid k\in\mathbb{Z}\}.$ It is important to point out that \(\langle g\...Recall that if $G$ is a group and $g\in G$ , then the cyclic subgroup generated by $g$ is given by $\langle g\rangle =\{g^k\mid k\in\mathbb{Z}\}.$ It is important to point out that $\langle g\rangle$ may be finite or infinite. If $G$ is a group and $g\in G$ such that $\langle g\rangle$ is a finite group, then the order of $g$ is the smallest positive integer $n$ such that $g^n=e$ .
3.2: Russell’s Paradox
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/03%3A_New_Page/3.02%3A_New_Page
In the 1920s, adjustments to Zermelo’s axioms were made by Abraham Fraenkel (1891–1965), Thoralf Skolem (1887–1963), and Zermelo that resulted in a collection of nine axioms, called ZFC, where ZF stan...In the 1920s, adjustments to Zermelo’s axioms were made by Abraham Fraenkel (1891–1965), Thoralf Skolem (1887–1963), and Zermelo that resulted in a collection of nine axioms, called ZFC, where ZF stands for Zermelo and Fraenkel and C stands for the Axiom of Choice, which is one of the nine axioms.
An Inquiry-Based Approach to Abstract Algebra (Ernst)
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)
This free and open-source textbook is designed to used with an inquiry-based learning (IBL) approach to a first-semester undergraduate abstract algebra course. The textbook starts with groups (up to t...This free and open-source textbook is designed to used with an inquiry-based learning (IBL) approach to a first-semester undergraduate abstract algebra course. The textbook starts with groups (up to the First Isomorphism Theorem) and finishes with an introduction to rings (up to quotients by maximal and prime ideals). While the textbook covers many of the standard topics, the focus is on building intuition and emphasizes visualization.
5: Cosets, Lagrange’s Theorem, and Normal Subgroups
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/05%3A_Cosets_Lagranges_Theorem_and_Normal_Subgroups
4.1: Introduction to Induction
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/04%3A_New_Page/4.01%3A_New_Page
In this chapter, we introduce mathematical induction, which is a proof technique that is useful for proving statements of the form (∀n ∈ $\mathbb{N}$ )P (n), or more generally (∀n ∈ $\mathbb{Z}$ )(n...In this chapter, we introduce mathematical induction, which is a proof technique that is useful for proving statements of the form (∀n ∈ $\mathbb{N}$ )P (n), or more generally (∀n ∈ $\mathbb{Z}$ )(n ≥ a =⇒ P (n)), where P (n) is some predicate and a ∈ $\mathbb{Z}$ .
5: The Real Numbers
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/05%3A_New_Page
6.1: Products of Groups
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/06%3A_Products_and_Quotients_of_Groups/6.01%3A_Products_of_Groups
Every finitely generated abelian group $G$ is isomorphic to a direct product of cyclic groups of the form \[\mathbb{Z}_{p_1^{n_1}}\times \mathbb{Z}_{p_2^{n_2}}\times \cdots \times \mathbb{Z}_{p_r^{n...Every finitely generated abelian group $G$ is isomorphic to a direct product of cyclic groups of the form $\mathbb{Z}_{p_1^{n_1}}\times \mathbb{Z}_{p_2^{n_2}}\times \cdots \times \mathbb{Z}_{p_r^{n_r}}\times \mathbb{Z}^k,$ where each $p_i$ is a prime number (not necessarily distinct).
2.5: More About Quantification
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_(Ernst)/02%3A_New_Page/2.05%3A_New_Page
Note that it would not be appropriate to utilize the “without loss of generality” approach to combine the two cases in the proof of Theorem 2.88 since the proof of the second case is not as simple as ...Note that it would not be appropriate to utilize the “without loss of generality” approach to combine the two cases in the proof of Theorem 2.88 since the proof of the second case is not as simple as swapping the roles of symbols in the proof of the first case.
1.2: What Should You Expect?
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/01%3A_Introduction/1.02%3A_What_Should_You_Expect
Up to this point, it is possible that your experience of mathematics has been about using formulas and algorithms. You are used to being asked to do things like: “solve for x", “take the derivative of...Up to this point, it is possible that your experience of mathematics has been about using formulas and algorithms. You are used to being asked to do things like: “solve for x", “take the derivative of this function", “integrate this function", etc. Your progress will be fueled by your ability to wrestle with mathematical ideas and to prove theorems. As you work through the book, you will find that you have ideas for proofs, but you are unsure of them.

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