Abstract Algebra: Group and Ring Theory

Vocabulary practice flashcards covering group isomorphism theorems, normal subgroups, ring properties, polynomial and matrix rings, ideal structures, Euclidean domains, PIDs, and UFDs.

First Isomorphism Theorem (for Groups)

If $\phi: G \rightarrow G'$ is a group homomorphism, then $\ker(\phi) \trianglelefteq G$, $\operatorname{Im}(\phi) \le G'$, and $G/\ker(\phi) \cong \operatorname{Im}(\phi)$.

Second Isomorphism Theorem (for Groups)

Let $G$ be a group with $A \le G$ and $N \trianglelefteq G$. Then $AN \le G$, $N \trianglelefteq AN$, $A \cap N \trianglelefteq A$, and $AN/N \cong A/(A \cap N)$.

Third Isomorphism Theorem (for Groups)

Let $G$ be a group with normal subgroups $A \trianglelefteq G$ and $B \trianglelefteq G$. If $A \subseteq B$, then $A \trianglelefteq B$, $B/A \trianglelefteq G/A$, and $(G/A)/(B/A) \cong G/B$.

Cayley's Theorem

Every group is isomorphic to a subgroup of a permutation group; specifically, for any group $G$, there exists a set $X$ and a subgroup $H \le S_X$ such that $G \cong H$.

Normal Subgroup

A subgroup $N$ of $G$ satisfying $xNx^{-1} \subseteq N$ for all $x \in G$ (equivalently, $xNx^{-1} = N$, $xN = Nx$, or $xy^{-1} \in N \iff y^{-1}x \in N$), denoted $N \trianglelefteq G$.

Quotient Group

The group of left cosets $G/N = \{xN \mid x \in G\}$ formed by a normal subgroup $N \trianglelefteq G$, where the binary operation is given by coset multiplication $(xN)(yN) = (xy)N$.

Kernel (Group Homomorphism)

For a group homomorphism $\phi: G \rightarrow G'$, the set of all elements in $G$ sent to the identity of $G'$, denoted $\ker(\phi) = \{x \in G \mid \phi(x) = 1\}$.

Image (Group Homomorphism)

For a group homomorphism $\phi: G \rightarrow G'$, the subgroup of $G'$ defined by $\operatorname{Im}(\phi) = \{\phi(x) \in G' \mid x \in G\}$.

Fiber

For a group homomorphism $\phi: G \rightarrow G'$ and an element $y \in \operatorname{Im}(\phi)$, the preimage set defined by $\phi^{-1}(y) = \{x \in G \mid \phi(x) = y\}$.

Ring

A set $R$ equipped with addition and multiplication such that $(R, +)$ is an Abelian group, multiplication is associative, and the distributive laws hold for all elements.

Commutative Ring

A ring $R$ in which multiplication is commutative, meaning $a \cdot b = b \cdot a$ for all $a, b \in R$.

Division Ring

A ring with identity $1 \neq 0$ in which every nonzero element has a multiplicative inverse.

Field

A commutative division ring; that is, a commutative ring with non-zero identity in which every non-zero element has a multiplicative inverse.

Zero Divisor

A nonzero element $a$ of a ring $R$ for which there exists a nonzero element $b \in R$ such that $ab = 0$ or $ba = 0$.

Unit (of a Ring)

An element $a$ in a ring $R$ with identity that has a multiplicative inverse, meaning there exists $b \in R$ such that $ab = ba = 1$.

Integral Domain

A commutative ring with identity that contains no zero divisors.

Subring

A subgroup $S$ of $(R, +)$ that is closed under the multiplication of $R$, so $a \cdot b \in S$ for all $a, b \in S$.

Center (of a Ring)

The subring of a ring $A$ consisting of elements $a \in A$ such that $ax = xa$ for all $x \in A$.

Ring Homomorphism

A map $\phi: R \rightarrow S$ between rings that preserves addition and multiplication: $\phi(a + b) = \phi(a) + \phi(b)$ and $\phi(ab) = \phi(a)\phi(b)$ for all $a, b \in R$.

Kernel (Ring Homomorphism)

For a ring homomorphism $\phi: R \rightarrow S$, the set of elements in $R$ mapped to the additive identity of $S$, given by $\operatorname{Ker}(\phi) = \{a \in R \mid \phi(a) = 0\}$.

Ideal (Two-Sided Ideal)

A subring $I$ of a ring $R$ that is closed under both left and right multiplication by elements of $R$, meaning $rI \subseteq I$ and $Ir \subseteq I$ for all $r \in R$.

Quotient Ring

The ring of cosets $R/I$ of an ideal $I \subseteq R$ under the operations $(r + I) + (s + I) = (r + s) + I$ and $(r + I)(s + I) = rs + I$.

First Isomorphism Theorem for Rings

If $\phi: R \rightarrow S$ is a ring homomorphism, then $\operatorname{Ker}(\phi)$ is an ideal of $R$, $\operatorname{Im}(\phi)$ is a subring of $S$, and $R/\operatorname{Ker}(\phi) \cong \operatorname{Im}(\phi)$.

Second Isomorphism Theorem for Rings

Let $R$ be a ring, $A$ a subring, and $B$ an ideal of $R$. Then $A + B$ is a subring of $R$, $A \cap B$ is an ideal of $A$, and $(A + B)/B \cong A/(A \cap B)$.

Third Isomorphism Theorem for Rings

Let $I$ and $J$ be ideals of a ring $R$ such that $I \subseteq J$. Then $J/I$ is an ideal of $R/I$ and $(R/I)/(J/I) \cong R/J$.

Fourth Isomorphism Theorem for Rings

For an ideal $I$ of $R$, the correspondence $A \mapsto A/I$ is an inclusion-preserving bijection between subrings (or ideals) of $R$ containing $I$ and subrings (or ideals) of $R/I$.

Polynomial Ring ($R[x]$)

The ring formed by infinite sequences $a = (a_0, a_1, a_2, \dots)$ in $R$ with $a_n = 0$ for almost all $n$, under component-wise addition and convolution multiplication $c_n = \sum_{i+j=n} a_i b_j$.

Indeterminate ($x$)

The specific sequence in $R[x]$ defined by $x = (0, 1, 0, 0, \dots)$, having $1$ in the second slot and $0$ everywhere else.

Degree (of a Polynomial)

For a nonzero polynomial $a(x) \in R[x]$, the largest integer $n$ such that $a_n \neq 0$, where $a_n$ is the leading coefficient.

General Linear Group ($\operatorname{GL}_n(R)$)

The group of units of the matrix ring $M_n(R)$, consisting of all invertible $n \times n$ matrices with entries in $R$.

Special Linear Group ($\operatorname{SL}_n(R)$)

The subgroup of $\operatorname{GL}_n(R)$ consisting of all $n \times n$ matrices with determinant equal to $1$.

Group Ring ($RG$)

For a commutative ring with identity $R$ and a finite group $G = \{g_1, \dots, g_n\}$, the ring of formal sums $RG = \{a_1 g_1 + \dots + a_n g_n \mid a_i \in R, g_i \in G\}$ with component-wise addition and multiplication $(a g_i)(b g_j) = (ab)(g_i g_j)$.

Augmentation Map

The ring homomorphism from the group ring $RG$ to $R$ defined by $\sum_{i=1}^n a_i g_i \mapsto \sum_{i=1}^n a_i$.

Ideal Generated by a Subset

For a subset $A \subseteq R$, the smallest ideal of $R$ containing $A$, denoted $(A)$, which is equal to the intersection of all ideals containing $A$.

Principal Ideal

An ideal generated by a single element, denoted $(a)$, which in a commutative ring with $1$ is given by $(a)=\left\lbrace{ar \mid r \in R}\right.\rbrace$ .

Prime Ideal

A proper ideal $P$ of a commutative ring $R$ such that whenever $ab \in P$, then $a \in P$ or $b \in P$ for any $a, b \in R$.

Maximal Ideal

A proper ideal $M$ of a ring $R$ such that if $I$ is an ideal of $R$ with $M \subseteq I \subseteq R$, then $M = I$ or $I = R$.

Norm (on an Integral Domain)

A map of sets $N: R \rightarrow \mathbb{N} \cup \{0\}$ on an integral domain $R$ such that for any nonzero $a, b \in R$, $N(a) \le N(ab)$.

Euclidean Domain

An integral domain $R$ equipped with a norm $N$ such that for any $a, b \in R$ with $b \neq 0$, there exist $q, r \in R$ such that $a = qb + r$, where $r = 0$ or $N(r) < N(b)$.

Euclidean Algorithm

A sequence of successive division steps in a Euclidean domain that terminates at the last nonzero remainder $r_n$, which satisfies $r_n = \gcd(a, b)$.

Greatest Common Divisor (in a Ring)

A nonzero element $d \in R$ such that $d \mid a$ and $d \mid b$, and if $c \mid a$ and $c \mid b$, then $c \mid d$.

Bezout Domain

An integral domain in which every ideal generated by two elements is principal.

Principal Ideal Domain (PID)

An integral domain in which every ideal is principal.

Ascending Chain Condition (ACC)

The condition on a ring $R$ that every chain of ideals $I_1 \subseteq I_2 \subseteq I_3 \subseteq \dots$ terminates, meaning there exists $N \in \mathbb{N}$ such that $I_n = I_{n+1}$ for all $n \ge N$.

Noetherian Ring

A ring that satisfies the ascending chain condition (ACC) on its ideals.

Prime Element

A nonzero element $p$ in an integral domain $R$ such that the principal ideal $(p)$ is a prime ideal, or equivalently, if $p \mid ab$, then $p \mid a$ or $p \mid b$.

Irreducible Element

A nonzero non-unit element $q$ in an integral domain $R$ such that whenever $q = ab$, either $a$ or $b$ is a unit in $R$.

Unique Factorization Domain (UFD)

An integral domain $D$ in which every nonzero non-unit element can be factored into a product of irreducible elements uniquely up to order and associate units.