Some properties of ideals in Cohen-Macaulay local rings

For a Cohen-Macaulay local ring $(R,\mathfrak{m})$ with canonical module, we study how relations between $\text{index}(R)$ and $\text{g}\mathrm{\ell }\mathrm{\ell }(R)$ and between $\text{index}(R)$ and $e(R)$ are preserved when factoring out regular sequences and localizing at prime ideals. We then give conditions for when ideals in a one-dimensional Cohen-Macaulay local ring are Elias and Burch, and use these conditions to study the relationship between Elias, Burch, and Ulrich ideals.