We calculate the degree of the algebra of covariants $\mathcal{C}_d$ for binary $d$-forms. We obtain the integral representation and asymptotic behavior of the degree.

Let $\mathbb{A}$ be a commutative $f$-algebra with unit. The sets of all ideals in $\mathbb{A}$ and all intersections of maximal ideals in $\mathbb{A}$ are denoted by $\mathfrak{I}(\mathbb{A})$ and $\mathfrak{IM}% (\mathbb{A})$, respectively. Whenever $\mathfrak{a}% \in\mathfrak{I}(\mathbb{A})$, we say that $\mathbb{A}$ is polynomially $\mathfrak{a}$-ideal if, for every $f\in\mathbb{A}$ with $p(f)\in\mathfrak{a}$ for some non-zero polynomial $p(x)$, there is an $f_{0}\in\mathfrak{a}$ such that $p(f+f_{0})=0$. We prove that if $\mathbb{A}$ is bounded inversion closed and $\mathfrak{a}\in\mathfrak{IM}(\mathbb{A})$, then $\mathbb{A}$ is polynomially $\mathfrak{a}$-ideal if and only if idempotents lift modulo $\mathfrak{a}$. This fact is based upon a systematic study of idempotent elements of an $f$-algebra. As a consequence, we show that, if $X$ is a Tychonoff space, then $C(X)$ is polynomially $\mathfrak{a}$-ideal for all $\mathfrak{a}\in\mathfrak{I}(C(X))$ if and only if $X$ is a $P$-space. Moreover, we prove that $C(X)$ is polynomially $\mathfrak{a}$-ideal for all $\mathfrak{a}\in\mathfrak{IM}(C(X))$ if and only if $X$ is strongly zero-dimensional. It turns out that this extends a theorem by Miers, namely, if $X$ is a compact Hausdorff space, then $C(X)$ is polynomially $\mathfrak{a}$-ideal for every uniformly closed ideal $\mathfrak{a}$ in $C(X)$ if and only if $X$ is totally disconnected.

This paper explores the relationship between real valued monomial valuations on $k(x,y)$, the resolution of cusp singularities and continued fractions. It is shown that, up to equivalence, there is a one-to-one correspondence between real valued monomial valuations on $k(x,y)$ and continued fraction expansions of real numbers between zero and one. This relationship with continued fractions is then used to provide a characterization of the valuation rings for real valued monomial valuations on $k(x,y)$. In the case when the monomial valuation is equivalent to an integral monomial valuation, we exhibit explicit generators of the valuation rings. Finally, we demonstrate that, if $\nu$ is a monomial valuation such that $\nu(x)=a$ and $\nu(y)=b$, where $a$ and $b$ are relatively prime positive integers larger than one, then $\nu$ governs a resolution of the singularities of the plane curve $x^{b}=y^{a}$ in a way we make explicit. Further, we provide an exact bound on the number of blow ups needed to resolve singularities in terms of the continued fraction of $a/b$.

Let $D$ be an integrally closed domain, $S(D)$ the set of star operations on $D$, $w$ the $w$-operation, and $S_w(D) = \{* \in S(D) \mid w \leq *\}$. Let $X$ be an indeterminate over $D$ and $N_v = \{f \in D[X] \mid c(f)_v = D\}$. In this paper, we show that, if $D$ is a Pr\"ufer $v$-multiplication domain (P$v$MD), then $|S_w(D)| = |S_w(D[X])| = |S(D[X]_{N_v})|$. We prove that $D$ is a P$v$MD if and only if $|\{* \in S_w(D) \mid *$ is of finite type$\}|\lt \infty$. We then use these results to give a complete characterization of integrally closed domains $D$ with $|S_w(D)| \lt \infty$.

In this paper, we determine the structure of the rings $\mathfrak{X}_{\Q}(H)$ of Fricke characters over $\Q$ of free abelian groups $H$ of rank $n \geq 1$. In particular, we consider the ideal $J$ in $\mathfrak{X}_{\Q}(H)$, generated by $\mbox{tr\,}x -2$ for any $x \in H$ and give a $\Q$-basis of each of the graded quotients $\mbox{gr}^k(J):=J^k/J^{k+1}$ for $k \geq 1$. Then we introduce a weight for each element of $\mathfrak{X}_{\Q}(H)$. By using the concept of this weight we show that $\mathfrak{X}_{\Q}(H)$ is an integral domain.

We examine the properties of certain mappings between the lattice $\mathcal{L}(R)$ of ideals of a commutative ring $R$ and the lattice $\mathcal{L}(_RM)$ of submodules of an $R$-module $M$, in particular considering when these mappings are lattice anti-homomorphisms. The mappings in question are the mapping $\alpha : \mathcal{L}(R) \rightarrow \mathcal{L}(_RM)$ defined by setting for each ideal $B$ of $R$, $\alpha(B)$ to be the submodule of $M$ consisting of all elements $m$ in $M$ with $Bm = 0$ and the mapping $\beta : \mathcal{L}(_RM) \rightarrow \mathcal{L}(R)$ defined by $\beta(N)$ is the annihilator in $R$ of $N$, for each submodule $N$ of $M$.