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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 $P\times\mathbb R$.
One example I could think of is the following: Define all operations just on the upper triangular part of the symmetric matrix and then just "define" the lower triangular part by mirroring the upper triangular part. This uses the idea that all information about a symmetric matrix is contained in its upper triangle
