Probabilistic questions have long been a topic of interest in finite group theory. In this work, we discuss the probability that two randomly chosen elements of a finite, commutative ring multiply to 0. In the first chapter, we will establish the basic properties of this probability, as well as the properties of annihilators and zero divisors. The next chapter will be dedicated to local rings and the probability of zero multiplication therein, with a subchapter about a special construction of local ring — idealization. In chapters three and four, we will find the best upper and lower bounds for the probability of zero multiplication in finite, commutative rings in general as well as in some rings with particular properties. We will also prove for which rings these bounds are sharp. Finally, we will determine all 12 non-trivial rings up to an isomorphism, whose probability of zero multiplication is greater than or equal to 3/8.
The majority of this thesis was based upon the findings in the article "The probability that the multiplication of two ring elements is zero" by the authors M. A. Esmkhani and S. M. Jafarian Amiri.
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