If something doesn’t behave like 0 in a ring or other algebraic structure then using that label is confusing and simply not done. You are free to use any symbol you want but mathematics is a human endeavor and as such communication is important. Using the symbol 0 signifies something to those with mathematical training. Zero can’t have an multiplicative inverse because anything you call 0 that has an multiplicative inverse makes it behave like something other than zero. So no one would use 0 to describe such an element. In a ring, or abelian group, the symbol 0 is reserved for the additive identity element.
Similarly, I could say snkwoo is what most people call a chair. A grammarian would say there is no word snkwoo even though I just defined it.
Your original comment was wrong and bad. Instead of just admitting it or moving on you’ve decided to double down and make another bad comment.
If something doesn’t behave like 0 in a ring or other algebraic structure then using that label is confusing and simply not done. You are free to use any symbol you want but mathematics is a human endeavor and as such communication is important. Using the symbol 0 signifies something to those with mathematical training. Zero can’t have an multiplicative inverse because anything you call 0 that has an multiplicative inverse makes it behave like something other than zero. So no one would use 0 to describe such an element. In a ring, or abelian group, the symbol 0 is reserved for the additive identity element.
Similarly, I could say snkwoo is what most people call a chair. A grammarian would say there is no word snkwoo even though I just defined it.
Your original comment was wrong and bad. Instead of just admitting it or moving on you’ve decided to double down and make another bad comment.