Hi, everybody. I'm recently reading W.Bruns and J.Herzog's famous book-Cohen-Macaulay Rings. I personally believe that it would be perfect if the authors provide for readers more concrete examples. After reading the first two sections of this book, I have two questions.


1. Given a non-negative integer n, how can we construct a non-Gorenstein Cohen-Macaulay ring with dimension n?
2. Except by the definitions of Cohen-Macaulay rings, is there a more efficient way to check the Cohen-Macaulayness of local rings?



Is there a simple method to test a local ring to be Cohen Macaulay?
Some interesting examples of Cohen-Macaulay but not Gorenstein rings:


1) Determinantal rings: Let $m\geq n\geq r>1$ be integers. Take $S=k[x\_{ij}]$ with $1\leq i\leq m, 1\leq j\leq n$ and $I$ be the ideal generated by all $r$ by $r$ minors. Then $R=S/I$ is CM, but is Gorenstein if and only if $m=n$. And $R$ has dimension $(m+n-r+1)(r-1)$.


2) Veronese subrings: Let $S=k[x\_1,\cdots,x\_n]$ and $R=S^{(d)}$ be the $k$-subalgebra of $S$ generated by the monomials of degree $d$. Then $R$ is always CM, but is Gorenstein if only if $d$ divides $n$. And $R$ has dimension $n$.


3) Semigroup rings: Let $R=k[t^{a\_1},\cdots, t^{a\_n}]$. $R$ is $1$-dimensional domain, so CM. $R$ is Gorenstein if and only if the semigroup generated by $(a\_1,\cdots,a\_n)$ is symmetric. For higher dimension one can use the trick in BCnrd's comment. 


