# Is there a way to give a ring structure on the group of symmetric matrices?

The set of $$n\times n$$ symmetric matrices with entries in $$\mathbb R$$ is an abelian group under matrix addition. However, it is not closed under matrix multiplication so one can't give it a ring structure using matrix multiplication.

Is there any other way to give it a non commutative ring structure?

I tried entry-wise multiplication (Hadamard Product) and also $$X\cdot Y:=XY+YX$$ but both of these are commutative operations and not very interesting to me.

When $$n\ge2$$, let $$N=\frac12n(n+1)\,(\ge3)$$ and let $$P\subset\mathbb R[x]$$ be the set of all polynomials of degree $$\le N-2$$. For any $$p=\sum_{k=0}^{N-2}a_kx^k$$ and $$q=\sum_{k=0}^{N-2}b_kx^k$$ in $$P$$, define $$p\cdot q=a_0q=\sum_{k=0}^{N-2}a_0b_kx^k$$. With this multiplication and the usual polynomial addition, $$P$$ becomes a non-commutative and non-unital ring ($$1$$ is not a unit element because $$p\cdot1\ne p$$ in general), and we may extend $$P$$ to a unital ring $$P\times\mathbb R$$ by defining an addition and a multiplication in a way akin to a Dorroh extension: \begin{aligned} (p,a)+(q,b)&=(p+q,a+b),\\ (p,a)\cdot(q,b)&=(p\cdot q+bp+aq,ab) \end{aligned} Now we can we pick a basis $$\{S_0,S_1,\ldots,S_{N-1}\}$$ of the vector space of all $$n\times n$$ real symmetric matrices and identify each symmetric matrix $$\sum_{k=0}^{N-1}a_kS_k$$ with the element $$\left(\sum_{k=0}^{N-2}a_kx^k,\,a_{N-1}\right)$$ in $$P\times\mathbb R$$.