UC Berkeley

This thesis explores two topics in commutative algebra. The first topic

is Betti tables, particularly of monomial ideals, and how these relate to

Betti tables of arbitrary graded ideals. We systematically study the

concept of mono, the largest monomial subideal of a given ideal, and

for an Artinian ideal I, deduce relations in the last column of the Betti

tables of I and mono(I). We then apply this philosophy towards a

conjecture of Postnikov-Shapiro, concerning Betti tables of certain ideals

generated by powers of linear forms: by studying monomial subideals

of the so-called power ideal, we deduce special cases of this conjecture.

The second topic concerns the group of units of a ring. Motivated by

the question of when a surjection of rings induces a surjection on unit

groups, we give a general sufficient condition for induced surjectivity to

hold, and introduce a new class of rings, called semi-fields, in the process.

As units are precisely the elements which avoid all the maximal ideals,

we then investigate infinite prime avoidance in general, and in this

direction, produce an example of a ring that is not a semi-field, for which

surjectivity on unit groups still holds.