What is a mothers ring?
Any ring that symbolizes each member of the family with a birth stone is a mothers ring. Here is a useful page that shows all of the birthstones used in mothers rings

Some people also like to include the birth stone of the mother and father as well as each child, and these are called family birth stone rings. Birth stone jewelry comes in many forms. There is even a variation called a daughters ring which always contains three birthstones – a large one in the center for the daughter and a smaller one on either side to represent the mother and father’s birth month.

There is a rich history of legend and symbolism behind family rings in general, and especially those that contain birth stones, such as gold mothers rings:

• A symbol of relationship:
  Since ancient times, rings have been recognized as a symbol of relationship, or a bond between the giver and the wearer of the ring. In some cultures, this was interpreted as a physical bond of ownership and control. For example in the ancient Greek myth of Prometheus, the god Zeus uses a ring of unbreakable adamantine to forever bind Prometheus to a rock as punishment for giving the secret of fire to mankind.
  More often than not, however, ancient cultures recognized the ring as a symbol of emotional attachment. The perfect circle of a gold mother ring represents continuity, as in the eternal return of the seasons. The shape of a circle has always been a metaphor for faithfulness, unity, eternity or perfection, and there are many examples of this in prehistoric rock art. Birth stone rings evolved from this tradition
  
  
• The magic of family birth stone rings:
  Gem stone rings and jewelry with enchanted powers have always played a large role in ancient folk tales. As the legends took hold, each gem became associated with specific powers, and the wearer of the gemstone was protected, or empowered accordingly. In addition, each sign of the zodiac, or birth month, was designated by a gemstone, which represented the personality traits of a child born under that sign.
  In a mystical sense, the woman who wears a personalized mothers ring is protected or empowered by the personality traits of each of her children. For example an amethyst (the February birth stone) is said to give emotional protection and peace of mind to the one who wears it. The personality attribute of amethyst is truthfulness. A sapphire (the September birth stone) is said to be the gem of heaven because it’s deep blue color was associated with the night sky. The personality trait of sapphire is innocence, and the wearer is empowered with purity of soul. Here is a useful page that summarizes all of the personality traits and legends associated with the gemstones in a mother’s ring
  
  
• The physical properties of birth stones:
  Science and spirituality often seem to cross paths, and the physics of gemstones are an exceptional example. Gemstones are among the most dense crystalline substances on the planet, formed from millions of years of pressure under the earth’s crust. The enchanting birth stone colors are the result of both the source element and the precise alignment of the crystals within the gem.
  In a physical sense, gemstones concentrate energy and redirect it along a certain prismatic path. The ancients believed that wearing a gem stone ring transferred the power of the gem from each finger to a different area of the body. The third finger of the left hand, for example, transferred energy to the heart which is why it is traditionally used for an engagement ring.
  The modern laser was invented with the use of a ruby, which concentrates ordinary light into a higher energy level that can be directed on a coherent path, a pinpoint that travels extreme distances without dispersing.
  Diamonds are the most highly conductive substance known, and are currently at the leading edge of future computer technology.
  If there is any physical reality that underlies the wearing of birth stones, it is perhaps summarized by the great psychologist Carl Jung, who said, "The meeting of two personalities is like the contact of two chemical substances; if there is any reaction, both are transformed."