Ring Theory
Notes for a public lecture on the occassion of the marriage of
Elizabeth Alice Cross and Andrew David Yager
11 December, 2004
Andrew and Bess, friends,
First, and most importantly - congratulations Andrew and Bess. We're delighted to be here, celebrating this day, and want you to know that you have our unreserved blessing as you begin your life together.
You know, today I'm having a little trouble with names. You see, it is hard to believe that twice in my lifetime, I could have witnessed a georgeous, intelligent, charming, generous and gracious girl, growing up in a distant place under a completely different name, only to reveal at the appropriate time that her name is Elizabeth Yager. 1 More than that, on both occassions, it seems that when the young man in question found her, he was able to describe her as living right next door. 2
I only have a minute or two, and unfortunately for the rest of us, I've realised that there are a few important things I need to tell Andrew to complete his education. Many of you know that I'm a maths lecturer, and some will know that I actually taught Andrew one course a couple of years ago - and looking back, I find that I didn't manage to cover a few important points in Ring Theory, and I need to rectify that.
Now rings are very important in Maths. You all know at least one example of a ring, the ordinary whole numbers, 0, ±1, ±2, etc. You can think of that, if you like, while I'm talking, but the point is that there are lots of different rings, and the ideas of ring theory apply equally to them all. The first thing to know about rings is that there are precisely two elements in a ring that are important enough to be mentioned in the definition. For the whole numbers, they are 0 and 1, but in other rings, they might have different names. Those elements are distinguished by the roles they play - their distinguishing property is that they preserve one another - in the whole numbers 0 preserves things when addition takes place, while 1 preserves things when multiplication takes place.
Which brings me to the things that go on in a ring. Rings are distinguished by the provision they make for addition and multiplication. You might wonder about subtraction, and the truth is that it is provided for - but viewed properly, as I've discovered again today, subtraction is really a special kind of addition. Division is another matter. While division can occur in some rings, the structure of a ring makes no provision for it at all. So whenever you are working in a ring, it is vital that you structure your arguments and draw your conclusions in a way that takes great care to avoid division.
Let me conclude the mathematical part of these few minutes by telling you that in a ring, it is not the elements themselves that are important at all. It's how they act together and the structures they form. The most important of these structures are the ideals of a ring - in fact, if you want to understand a ring in any depth, it is the ideals that you should study. Amongst these ideals, the most important ones are the principal ideals, the ideals that a generated by a single element. Every ring has many of these, and Andrew and Bess, I hope that the principal ideals of your ring will be generated by the things that have been the focus of the ceremony today; love, trust, understanding, faithfulness, mutual forgiveness, and above all, your shared faith in Jesus Christ.
Today's a special day. In a real sense, Elizabeth (that's my Elizabeth) and I have been looking forward and preparing this day for all of Andrew's life. We realized from the beginning that God had entrusted him to us for a limited time, and that our job was to prepare him for life, educate him, help him to grow in maturity and learn to accept responsibility, to introduce him to our Saviour, Jesus and, after investing all that we could in him, to send him out into the world. We knew that God had not designed him to face that task on his own, and so we've been praying for the partner he would find to share life with, and we share Andrew's joy at finding that partner in Bess.
Andrew, we've witnessed something as significant as your birth here today. A new family has been born, and your mum and I want you to know that, for you, your new family must come first. Be fiercely loyal to Bess, put her first in everything, listen to her and do all you can to understand her dreams, her hopes and her needs. She is very precious, the best gift you will ever receive. We are delighted with the young man you have grown into, and we know that as you work together to nuture and mature the marriage that was born today, we will continue to delight in your happiness. And you both should know by now can rely on us to be there to support you in the tough times that undoubtedly you will face from time to time. May God bless you richly in every aspect of your life together.
1 My wife was named Elizabeth May Barfoot by her parents, but was known by everyone as Elsie Barfoot. Andrew's wife was named Elizabeth Alice Cross by her parents, but was known to everyone as Bess Cross.
2 In 1979, I was working towards my PhD in Paris, and spent the summer holidays with my great aunt and her family near Magherafelt, Northern Ireland. Elsie Barfoot lived next door! Bess Cross lived in Avoca on the NSW Central Coast, about 140km from our home, which is about 10km from Macquarie University. However, Andrew was quite happy to tell just about anyone that the natural route from the University to our home involved a 280km journey.