Definition Punctuation

docs/math/abstract-algebra/16-rings.md

@@ -56,7 +56,7 @@ Note that some books impose the condition that $1 \neq 0$. If $1 = 0$, we can sh
 
 ## Section 16.3 - Ring Homomorphisms and Ideals
 
-**Definition** Given rings $R$ and $S$, and a mapping $\varphi: R \rightarrow S$, we say that $\varphi$ is a *ring homomorphism* if the following are satisfied for all elements of $R$:
+**Definition**. Given rings $R$ and $S$, and a mapping $\varphi: R \rightarrow S$, we say that $\varphi$ is a *ring homomorphism* if the following are satisfied for all elements of $R$:
 
 $$
 \begin{align}

docs/math/abstract-algebra/17-polynomial-rings.md

@@ -46,7 +46,7 @@ where either $\deg r(x) < \deg g(x)$ or $r(x)$ is the zero polynomial.
 
 ## Section 17.3 Irreducible Polynomials
 
-**Definition** A non-constant polynomial $f(x) \ in F[x]$ is *irreducible* over a field $F$ if it cannot be expressed as the product of two non-identity polynomials $g(x)$ and $h(x)$ in $F[x]$, with the degree of both polynomials strictly less than the  degree of $f(x)$.
+**Definition**. A non-constant polynomial $f(x) \ in F[x]$ is *irreducible* over a field $F$ if it cannot be expressed as the product of two non-identity polynomials $g(x)$ and $h(x)$ in $F[x]$, with the degree of both polynomials strictly less than the  degree of $f(x)$.
 
 **Lemma**. Let $p(x) \in \mathbb{Q}[x]$. Then, with $r, s \in \mathbb{Z}, a(x) \in \mathbb{N}[x]$, we can write $p(x) = \frac{r}{s} a(x)$.
 

docs/math/abstract-algebra/18-integral-domains.md

@@ -57,7 +57,7 @@ Additionally, $F_D$ is unique. That is, given field $E$ such that $E \supset D$,
 
 **Theorem**. 18.15: Every PID is a UFD. Note that the converse is not true.
 
-**Corollary** 18.16: Let $F$ be a field. Then, $F[x]$ is a UFD.
+**Corollary**. 18.16: Let $F$ be a field. Then, $F[x]$ is a UFD.
 
 ---
 

docs/math/abstract-algebra/DF-10-modules.md

@@ -15,7 +15,7 @@
 
 **Remark**. Modules over a field $F$ and vector spaces over $F$ are identical.
 
-**Definition** An *R-submodule* is a subset$N \subseteq M$ which is closed under the action taken forall $r \in R$. That is, given $r \in R, n \in N$, then $rn \in N$. Every module has at least two submodules: itself and the trivial (empty) submodule.
+**Definition**. An *R-submodule* is a subset$N \subseteq M$ which is closed under the action taken forall $r \in R$. That is, given $r \in R, n \in N$, then $rn \in N$. Every module has at least two submodules: itself and the trivial (empty) submodule.
 
 **Remark**. If $F$ is a field, submodules are equivalent to subspaces.
 
@@ -124,7 +124,7 @@ This direct product is in itself an $R$-module.
 
 **Corollary**. If $F$ is a free $R$-module with basis $A$, then $F \cong F(A)$.
 
-**Definition** For a free module $F$ with basis $A$, if $R$ is commutative, then the *rank* of $F$ is the cardinality of $A$.
+**Definition**. For a free module $F$ with basis $A$, if $R$ is commutative, then the *rank* of $F$ is the cardinality of $A$.
 
 ## Section 10.4 - Tensor Products of Modules
 

docs/math/abstract-algebra/DF-12-modules-pids.md

@@ -30,7 +30,7 @@ $$
 
 This is the *torsion submodule* of $M$. If $\text{Tor}(M)$ is empty, then $M$ is *torsion-free*.
 
-**Definition** Let $R$ be an integral domain and $M$ be an $R$-module. Then, given a submodule $N$,
+**Definition**. Let $R$ be an integral domain and $M$ be an $R$-module. Then, given a submodule $N$,
 
 $$
 \text{Ann}_R(N) = \{r \in R | rn = 0 \text{ for all } n \in N \}

docs/math/diffeq/2-1st-order.md

@@ -65,7 +65,7 @@ $$
 \frac{dy}{dt} + p(t)y = g(t)
 $$
 
-To find a solution to this differential equation, construct the **integrating factor** $\mu(t)$.
+To find a solution to this differential equation, construct the **integrating factor**. $\mu(t)$.
 
 $$\mu(t) = k \exp(\int p(t) dt)$$
 

docs/math/real-analysis/2-reals.md

@@ -21,7 +21,7 @@
 2. $\mathbb{N} \subseteq F^+$
 3. If $a \in F^+$, then $\frac{1}{a} \in F^+$
 
-**Definition** The order relation $a > b$ and $b < a$ is defined by $a - b \in F^+$.
+**Definition**. The order relation $a > b$ and $b < a$ is defined by $a - b \in F^+$.
 
 **Theorem**. If $a, b, c \in F$, then
 
@@ -76,9 +76,9 @@
 2. $|a - b| = |b - a|$
 3. $|a - c| \leq |a - b| + |b + c|$
 
-**Definition** A set together with a function satisfying these three properties is known as a *metric space*.
+**Definition**. A set together with a function satisfying these three properties is known as a *metric space*.
 
-**Definition** The $\varepsilon$-neighborhood of $a \in \mathbb{R}$, denoted $V_\varepsilon(a)$ is the set of all real numbers $x \in \mathbb{R}$ such that $|x - a| < \varepsilon$. That is,
+**Definition**. The $\varepsilon$-neighborhood of $a \in \mathbb{R}$, denoted $V_\varepsilon(a)$ is the set of all real numbers $x \in \mathbb{R}$ such that $|x - a| < \varepsilon$. That is,
 
 $$
 V_\varepsilon(a) = (a - \varepsilon, a + \varepsilon)