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I am not sure if this is appropriate for MO. If not, I shall be happy to take it to SE. For a local ring $(R,m)$, given any proper ideal $I$, the (Krull) dimension (from here on dimension means Krull …

This was one of the aspects of algebra that I enjoyed the most while first learning it. This is the way I would develop the subject: 1) Introduction to exact sequences with an emphasis on short exact …

Let $R$ be a Noetherian ring, and let $I$ be an ideal of $R$. Fix a rational number $a=\frac{p}{q}$ with $p, q\in \mathbb{Z_\geq 0}$ $q\neq 0$. We define $I_a = \{x \in R: x^q\in \overline{I^p}\}$, wh …

This is with reference to theorem 5.20 in Vasconcelos book linked (google books) here: http://tinyurl.com/2967eov I shall restate the theorem here for easy reference: "If $A=k[[x_1,x_2,...,x_n]]/I$ i …

Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. An ideal (that can be) generated by monomials is called a monomial ideal. For the monomial ideal $M=(m_1,m_2,. …

Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. A monomial in $R$ is an element which is product (with repetitions allowed) of the indeterminates. Monomial id …

The following is a corollary of the Briançon-Skoda theorem: If $R$ is a regular Noetherian ring of Krull dimension $d$ and $f_1,f_2,...,f_{d+1}\in R$. Then, $f_1^df_2^d...f_{d+1}^d \in (f_1^{d+1},f_2 …

If $R=k[x_1,...,x_n]$ is a polynomial ring in $n$ indeterminates over a field $k$ of characteristic $0$, there is a characterization of the symbolic square of a prime ideal $P$ (the $n$-th symbolic po …

Given a toric ideal, say $J$, in a polynomial ring $k[x_1,...,x_n]$ we can find a finite generating set for $J$. Is it possible, perhaps under additional assumptions on the structure of $J$, to give …

I think this is certainly true if $p$ is generated by a regular sequence. In this case, $R/p$ is regular, local and hence Cohen Macaulay. Hence, $R/p$ is a maximal Cohen Macaulay module over $R/Ann(R/ …

I am unable to write this is in comments. While this is not an answer to your question, a similar structure theorem holds for Principal ideal rings where every finitely generated module is isomorphic …