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• CommentRowNumber1.
• CommentAuthorColin Tan
• CommentTimeApr 29th 2014

Currently, an element x in a nonassociative algebra A is nilpotent if there exist a natural number n such that $x^n = 0$.

I want to say that a nilpotent left ideal of a ring R is a nilpotent element in the set of left ideals of R. To say that, I have to determine the structure of the set of left ideals of a ring under addition and multiplication. Wikipedia says that the set of ideals of a ring is a complete modular lattice. Is a complete modular lattice a nonassociative algebra? If not, do people talk about nilpotent elements in a lattice?

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeApr 29th 2014
• (edited Apr 30th 2014)

A lattice is a particular kind of poset, one that has binary joins and binary meets; a complete lattice has arbitrary joins and arbitrary meets. Modularity is a certain Horn condition that holds for certain types of lattice; very typically they are lattices of congruence relations for certain types of algebraic theories called Mal’cev theories.

Inasmuch as joins are coproducts and meets are (cartesian) products, sometimes one might see algebraic notation that could suggest ring-like behavior in a lattice, and this is reasonable for certain types of lattices like Boolean algebras, but for the lattice of ideals of a ring, it is relatively rare that meets distribute over joins (and the join is never, or essentially never, an abelian group multiplication). In fact the join of two (say left) ideals $I$, $J$ turns out to be $I + J$ (the ideal consisting of all elements $i + j$ where $i \in I$ and $j \in J$); it’s the smallest ideal that contains the union $I \cup J$. The meet of two ideals is their intersection $I \cap J$. Then, for ideals of a ring, there is a third operation $I \cdot J$ whose elements consist of products $i \cdot j$; this operation is not specifically a lattice-theoretic concept, but is special to ideals of a ring – and this is the operation that is meant when one refers to nilpotency.

So, to answer the question: the specifically lattice-theoretic operations of meet and join sometimes (but not all that often, in the scheme of things) form a rig, if distributivity holds, but always a commutative rig. Using the special operation $\cdot$ for multiplication instead of meet, the lattice of ideals forms a rig, but always an associative rig. Basically no one thinks of a complete modular lattice per se as forming a nonassociative algebra.

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeApr 30th 2014
• (edited Apr 30th 2014)

Colin, I see you are taking a more active hand in edits. That’s fine and generally welcome, but in accordance with established custom (see the very first paragraph here), please leave a note at the nForum the next time you make more than trivial edits. I am looking particularly at your edits to smooth topos. While the edits are mostly not deep ones, one of the main purposes of the nForum is to discuss such matters openly; this also helps avoid situations where an author discovers by accident that something he thought was clearly and thoughtfully worded has been replaced by something rather less to his liking.

The way our little culture has developed, nLab authors try to avoid undoing or erasing the competent work of others in favor of one’s own preferred formulations; the general idea is always to add, and try avoid subtracting unless one is prepared to argue that a definite improvement was made (which might be, e.g., smoothing over an infelicity of the English of a non-native speaker, or something that seemed ambiguous).

I went a step further and numbered the definitions, and capitalized the first letters of subsection titles; I also changed back to include the full phrase “amazing right adjoint”.

• CommentRowNumber4.
• CommentAuthorTobyBartels
• CommentTimeApr 30th 2014

I've just edited nilpotent element to allow the context to be a (possibly nonassociative) rig. The left ideals of a ring form an associative rig under addition and multiplication, as Todd said, so you can use this context directly.

• CommentRowNumber5.
• CommentAuthorColin Tan
• CommentTimeMay 8th 2014
Todd, I read your message and will learn to follow the editing customs of the Lab. I enjoy editing and learning from you all. If any of my editing has been against the accepted norms, then do revert them.